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Theorem nodense 27063
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 27060 . 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 27019 . . . . 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 5963 . . . . 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 27059 . . . . . . . 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 27048 . . . . . . . . . . 11 bday : No –ontoβ†’On
7 fof 6760 . . . . . . . . . . 11 ( bday : No –ontoβ†’On β†’ bday : No ⟢On)
86, 7ax-mp 5 . . . . . . . . . 10 bday : No ⟢On
9 0elon 6375 . . . . . . . . . 10 βˆ… ∈ On
108, 9f0cli 7052 . . . . . . . . 9 ( bday β€˜π΄) ∈ On
1110onelssi 6436 . . . . . . . 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 27019 . . . . . . . 8 (𝐴 ∈ No β†’ ( bday β€˜π΄) = dom 𝐴)
1413ad2antrr 725 . . . . . . 7 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ ( bday β€˜π΄) = dom 𝐴)
1512, 14sseqtrd 3988 . . . . . 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 3931 . . . . . 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 2785 . . . 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 2773 . . 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 2834 . 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 27058 . . . . 5 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ 𝐴 <s 𝐡) β†’ ∩ {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)} ∈ On)
2221adantrl 715 . . . 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 27062 . . . . . . . . . . . . 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 1122 . . . . . . . . . . 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 420 . . . . . . . . . 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 496 . . . . . . . . 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 2733 . . . . . . . . 9 βˆ… = βˆ…
2927, 28jctir 522 . . . . . . . 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 1337 . . . . . . 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 6859 . . . . . . . 8 (π΄β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ V
32 0ex 5268 . . . . . . . 8 βˆ… ∈ V
3331, 32brtp 5484 . . . . . . 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 6850 . . . . . . 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 27049 . . . . . . . 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 2775 . . . . . 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 5137 . . . . 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 6865 . . . . . . 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 2739 . . . . . 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 521 . . . 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 3308 . . . . . 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 6846 . . . . . . 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 6846 . . . . . . 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 5122 . . . . . 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 632 . . . . 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 3583 . . . 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 585 . . 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 766 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ 𝐴 ∈ No )
52 sltval 27018 . . . 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 585 . . 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 257 . 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 483 . . . . . 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 27061 . . . . . . 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 408 . . . . . 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 2775 . . . . 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 3140 . . . 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 497 . . . . . . . 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 521 . . . . . . 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 1339 . . . . . 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 6859 . . . . . . 7 (π΅β€˜βˆ© {π‘Ž ∈ On ∣ (π΄β€˜π‘Ž) β‰  (π΅β€˜π‘Ž)}) ∈ V
6432, 63brtp 5484 . . . . . 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 5131 . . . 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 3308 . . . . . 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 6846 . . . . . . 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 5122 . . . . . 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 632 . . . . 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 3583 . . . 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 836 . . 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 768 . . . 4 (((𝐴 ∈ No ∧ 𝐡 ∈ No ) ∧ (( bday β€˜π΄) = ( bday β€˜π΅) ∧ 𝐴 <s 𝐡)) β†’ 𝐡 ∈ No )
74 sltval 27018 . . . 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 585 . . 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 257 . 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 6846 . . . . 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 2819 . . . 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 5113 . . . 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 5112 . . . 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 1437 . . 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 3583 . 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 1373 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 397   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  {ctp 4594  βŸ¨cop 4596  βˆ© cint 4911   class class class wbr 5109  dom cdm 5637   β†Ύ cres 5639  Oncon0 6321  βŸΆwf 6496  β€“ontoβ†’wfo 6498  β€˜cfv 6500  1oc1o 8409  2oc2o 8410   No csur 27011   <s cslt 27012   bday cbday 27013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1o 8416  df-2o 8417  df-no 27014  df-slt 27015  df-bday 27016
This theorem is referenced by:  nocvxminlem  27146  addsproplem6  27315  negsproplem6  27360
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