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Theorem nodense 27192
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 27189 . 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 27148 . . . . 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 6003 . . . . 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 27188 . . . . . . . 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 27177 . . . . . . . . . . 11 bday : No –ontoβ†’On
7 fof 6805 . . . . . . . . . . 11 ( bday : No –ontoβ†’On β†’ bday : No ⟢On)
86, 7ax-mp 5 . . . . . . . . . 10 bday : No ⟢On
9 0elon 6418 . . . . . . . . . 10 βˆ… ∈ On
108, 9f0cli 7099 . . . . . . . . 9 ( bday β€˜π΄) ∈ On
1110onelssi 6479 . . . . . . . 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 27148 . . . . . . . 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 4022 . . . . . 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 3965 . . . . . 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 2784 . . . 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 2772 . . 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 2833 . 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 27187 . . . . 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 27191 . . . . . . . . . . . . 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 1121 . . . . . . . . . . 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 419 . . . . . . . . . 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 495 . . . . . . . . 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 2732 . . . . . . . . 9 βˆ… = βˆ…
2927, 28jctir 521 . . . . . . . 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 1336 . . . . . . 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 6904 . . . . . . . 8 (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ V
32 0ex 5307 . . . . . . . 8 βˆ… ∈ V
3331, 32brtp 5523 . . . . . . 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 6895 . . . . . . 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 27178 . . . . . . . 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 2774 . . . . . 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 5176 . . . . 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 6910 . . . . . . 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 2738 . . . . . 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 3063 . . . . 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 520 . . . 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 3322 . . . . . 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 6891 . . . . . . 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 6891 . . . . . . 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 5161 . . . . . 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 631 . . . . 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 3612 . . . 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 584 . . 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 27147 . . . 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 584 . . 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 482 . . . . . 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 27190 . . . . . . 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 407 . . . . . 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 2774 . . . . 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 3146 . . . 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 496 . . . . . . . 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 520 . . . . . . 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 1338 . . . . . 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 6904 . . . . . . 7 (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ V
6432, 63brtp 5523 . . . . . 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 5170 . . . 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 3322 . . . . . 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 6891 . . . . . . 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 5161 . . . . . 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 631 . . . . 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 3612 . . . 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 27147 . . . 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 584 . . 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 6891 . . . . 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 2818 . . . 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 5152 . . . 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 5151 . . . 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 1436 . . 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 3612 . 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 1372 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 396   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {ctp 4632  βŸ¨cop 4634  βˆ© cint 4950   class class class wbr 5148  dom cdm 5676   β†Ύ cres 5678  Oncon0 6364  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€˜cfv 6543  1oc1o 8458  2oc2o 8459   No csur 27140   <s cslt 27141   bday cbday 27142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1o 8465  df-2o 8466  df-no 27143  df-slt 27144  df-bday 27145
This theorem is referenced by:  nocvxminlem  27276  addsproplem6  27455  negsproplem6  27504  mulsproplem13  27581  mulsproplem14  27582
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