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Theorem nodense 27643
Description: Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Axiom SD of [Alling] p. 184. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodense (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ βˆƒπ‘₯ ∈ No (( bday β€˜π‘₯) ∈ ( bday β€˜π΄) ∧ 𝐴 <s π‘₯ ∧ π‘₯ <s 𝐡))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡

Proof of Theorem nodense
Dummy variables π‘Ž 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nodenselem6 27640 . 2 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ No )
2 bdayval 27599 . . . . 5 ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ No β†’ ( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})) = dom (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
31, 2syl 17 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})) = dom (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
4 dmres 6019 . . . . 5 dom (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = (∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∩ dom 𝐴)
5 nodenselem5 27639 . . . . . . . 8 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∈ ( bday β€˜π΄))
6 bdayfo 27628 . . . . . . . . . . 11 bday : No –ontoβ†’On
7 fof 6814 . . . . . . . . . . 11 ( bday : No –ontoβ†’On β†’ bday : No ⟢On)
86, 7ax-mp 5 . . . . . . . . . 10 bday : No ⟢On
9 0elon 6426 . . . . . . . . . 10 βˆ… ∈ On
108, 9f0cli 7111 . . . . . . . . 9 ( bday β€˜π΄) ∈ On
1110onelssi 6487 . . . . . . . 8 (∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∈ ( bday β€˜π΄) β†’ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} βŠ† ( bday β€˜π΄))
125, 11syl 17 . . . . . . 7 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} βŠ† ( bday β€˜π΄))
13 bdayval 27599 . . . . . . . 8 (𝐴 ∈ No β†’ ( bday β€˜π΄) = dom 𝐴)
1413ad2antrr 724 . . . . . . 7 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ( bday β€˜π΄) = dom 𝐴)
1512, 14sseqtrd 4020 . . . . . 6 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} βŠ† dom 𝐴)
16 df-ss 3964 . . . . . 6 (∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} βŠ† dom 𝐴 ↔ (∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∩ dom 𝐴) = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})
1715, 16sylib 217 . . . . 5 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∩ dom 𝐴) = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})
184, 17eqtrid 2779 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ dom (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})
193, 18eqtrd 2767 . . 3 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})) = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})
2019, 5eqeltrd 2828 . 2 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})) ∈ ( bday β€˜π΄))
21 nodenselem4 27638 . . . . 5 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ 𝐴 <s 𝐡) β†’ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∈ On)
2221adantrl 714 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∈ On)
23 nodenselem8 27642 . . . . . . . . . . . . 13 ((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ ( bday β€˜π΄) = ( bday β€˜π΅)) β†’ (𝐴 <s 𝐡 ↔ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o)))
2423biimpd 228 . . . . . . . . . . . 12 ((𝐴 ∈ No ∧ 𝐡 ∈ No ∧ ( bday β€˜π΄) = ( bday β€˜π΅)) β†’ (𝐴 <s 𝐡 β†’ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o)))
25243expia 1118 . . . . . . . . . . 11 ((𝐴 ∈ No ∧ 𝐡 ∈ No ) β†’ (( bday β€˜π΄) = ( bday β€˜π΅) β†’ (𝐴 <s 𝐡 β†’ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o))))
2625imp32 417 . . . . . . . . . 10 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o))
2726simpld 493 . . . . . . . . 9 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o)
28 eqid 2727 . . . . . . . . 9 βˆ… = βˆ…
2927, 28jctir 519 . . . . . . . 8 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ βˆ… = βˆ…))
30293mix1d 1333 . . . . . . 7 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ βˆ… = βˆ…) ∨ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ βˆ… = 2o) ∨ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = βˆ… ∧ βˆ… = 2o)))
31 fvex 6913 . . . . . . . 8 (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ V
32 0ex 5309 . . . . . . . 8 βˆ… ∈ V
3331, 32brtp 5527 . . . . . . 7 ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩}βˆ… ↔ (((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ βˆ… = βˆ…) ∨ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 1o ∧ βˆ… = 2o) ∨ ((π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = βˆ… ∧ βˆ… = 2o)))
3430, 33sylibr 233 . . . . . 6 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩}βˆ…)
3519fveq2d 6904 . . . . . . 7 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
36 fvnobday 27629 . . . . . . . 8 ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ No β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))) = βˆ…)
371, 36syl 17 . . . . . . 7 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))) = βˆ…)
3835, 37eqtr3d 2769 . . . . . 6 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = βˆ…)
3934, 38breqtrrd 5178 . . . . 5 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
40 fvres 6919 . . . . . . 7 (𝑦 ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΄β€˜π‘¦))
4140eqcomd 2733 . . . . . 6 (𝑦 ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦))
4241rgen 3059 . . . . 5 βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦)
4339, 42jctil 518 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ∧ (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})))
44 raleq 3318 . . . . . 6 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ (βˆ€π‘¦ ∈ π‘₯ (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ↔ βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦)))
45 fveq2 6900 . . . . . . 7 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ (π΄β€˜π‘₯) = (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
46 fveq2 6900 . . . . . . 7 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
4745, 46breq12d 5163 . . . . . 6 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ ((π΄β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯) ↔ (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})))
4844, 47anbi12d 630 . . . . 5 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ ((βˆ€π‘¦ ∈ π‘₯ (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ∧ (π΄β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯)) ↔ (βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ∧ (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))))
4948rspcev 3609 . . . 4 ((∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∈ On ∧ (βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ∧ (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))) β†’ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ∧ (π΄β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯)))
5022, 43, 49syl2anc 582 . . 3 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ∧ (π΄β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯)))
51 simpll 765 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ 𝐴 ∈ No )
52 sltval 27598 . . . 4 ((𝐴 ∈ No ∧ (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ No ) β†’ (𝐴 <s (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ↔ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ∧ (π΄β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯))))
5351, 1, 52syl2anc 582 . . 3 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (𝐴 <s (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ↔ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) ∧ (π΄β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯))))
5450, 53mpbird 256 . 2 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ 𝐴 <s (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
5541adantl 480 . . . . . 6 ((((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) ∧ 𝑦 ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) β†’ (π΄β€˜π‘¦) = ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦))
56 nodenselem7 27641 . . . . . . 7 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (𝑦 ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ (π΄β€˜π‘¦) = (π΅β€˜π‘¦)))
5756imp 405 . . . . . 6 ((((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) ∧ 𝑦 ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) β†’ (π΄β€˜π‘¦) = (π΅β€˜π‘¦))
5855, 57eqtr3d 2769 . . . . 5 ((((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) ∧ 𝑦 ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦))
5958ralrimiva 3142 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦))
6026simprd 494 . . . . . . . 8 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o)
6160, 28jctil 518 . . . . . . 7 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (βˆ… = βˆ… ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o))
62613mix3d 1335 . . . . . 6 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ((βˆ… = 1o ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = βˆ…) ∨ (βˆ… = 1o ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o) ∨ (βˆ… = βˆ… ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o)))
63 fvex 6913 . . . . . . 7 (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ V
6432, 63brtp 5527 . . . . . 6 (βˆ…{⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ↔ ((βˆ… = 1o ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = βˆ…) ∨ (βˆ… = 1o ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o) ∨ (βˆ… = βˆ… ∧ (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) = 2o)))
6562, 64sylibr 233 . . . . 5 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ βˆ…{⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
6638, 65eqbrtrd 5172 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
67 raleq 3318 . . . . . 6 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ (βˆ€π‘¦ ∈ π‘₯ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦) ↔ βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦)))
68 fveq2 6900 . . . . . . 7 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ (π΅β€˜π‘₯) = (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))
6946, 68breq12d 5163 . . . . . 6 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ (((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜π‘₯) ↔ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})))
7067, 69anbi12d 630 . . . . 5 (π‘₯ = ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} β†’ ((βˆ€π‘¦ ∈ π‘₯ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦) ∧ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜π‘₯)) ↔ (βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦) ∧ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))))
7170rspcev 3609 . . . 4 ((∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∈ On ∧ (βˆ€π‘¦ ∈ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦) ∧ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}))) β†’ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦) ∧ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜π‘₯)))
7222, 59, 66, 71syl12anc 835 . . 3 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦) ∧ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜π‘₯)))
73 simplr 767 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ 𝐡 ∈ No )
74 sltval 27598 . . . 4 (((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ No ∧ 𝐡 ∈ No ) β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) <s 𝐡 ↔ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦) ∧ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜π‘₯))))
751, 73, 74syl2anc 582 . . 3 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) <s 𝐡 ↔ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘¦) = (π΅β€˜π‘¦) ∧ ((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})β€˜π‘₯){⟨1o, βˆ…βŸ©, ⟨1o, 2o⟩, βŸ¨βˆ…, 2o⟩} (π΅β€˜π‘₯))))
7672, 75mpbird 256 . 2 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) <s 𝐡)
77 fveq2 6900 . . . . 5 (π‘₯ = (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) β†’ ( bday β€˜π‘₯) = ( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})))
7877eleq1d 2813 . . . 4 (π‘₯ = (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) β†’ (( bday β€˜π‘₯) ∈ ( bday β€˜π΄) ↔ ( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})) ∈ ( bday β€˜π΄)))
79 breq2 5154 . . . 4 (π‘₯ = (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) β†’ (𝐴 <s π‘₯ ↔ 𝐴 <s (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})))
80 breq1 5153 . . . 4 (π‘₯ = (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) β†’ (π‘₯ <s 𝐡 ↔ (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) <s 𝐡))
8178, 79, 803anbi123d 1432 . . 3 (π‘₯ = (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) β†’ ((( bday β€˜π‘₯) ∈ ( bday β€˜π΄) ∧ 𝐴 <s π‘₯ ∧ π‘₯ <s 𝐡) ↔ (( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})) ∈ ( bday β€˜π΄) ∧ 𝐴 <s (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∧ (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) <s 𝐡)))
8281rspcev 3609 . 2 (((𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ No ∧ (( bday β€˜(𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)})) ∈ ( bday β€˜π΄) ∧ 𝐴 <s (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∧ (𝐴 β†Ύ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) <s 𝐡)) β†’ βˆƒπ‘₯ ∈ No (( bday β€˜π‘₯) ∈ ( bday β€˜π΄) ∧ 𝐴 <s π‘₯ ∧ π‘₯ <s 𝐡))
831, 20, 54, 76, 82syl13anc 1369 1 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ βˆƒπ‘₯ ∈ No (( bday β€˜π‘₯) ∈ ( bday β€˜π΄) ∧ 𝐴 <s π‘₯ ∧ π‘₯ <s 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ w3o 1083   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2936  βˆ€wral 3057  βˆƒwrex 3066  {crab 3428   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4324  {ctp 4634  βŸ¨cop 4636  βˆ© cint 4951   class class class wbr 5150  dom cdm 5680   β†Ύ cres 5682  Oncon0 6372  βŸΆwf 6547  β€“ontoβ†’wfo 6549  β€˜cfv 6551  1oc1o 8484  2oc2o 8485   No csur 27591   <s cslt 27592   bday cbday 27593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-1o 8491  df-2o 8492  df-no 27594  df-slt 27595  df-bday 27596
This theorem is referenced by:  nocvxminlem  27728  addsproplem6  27909  negsproplem6  27963  mulsproplem13  28046  mulsproplem14  28047
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