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Theorem nocvxminlem 33538
 Description: Lemma for nocvxmin 33539. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
Assertion
Ref Expression
nocvxminlem ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ¬ 𝑋 <s 𝑌))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦,𝑌,𝑧
Allowed substitution hint:   𝑌(𝑥)

Proof of Theorem nocvxminlem
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 breq1 5036 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 <s 𝑧𝑋 <s 𝑧))
21anbi1d 633 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑦)))
32imbi1d 346 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
43ralbidv 3127 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
5 breq2 5037 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑧 <s 𝑦𝑧 <s 𝑌))
65anbi2d 632 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑋 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑌)))
76imbi1d 346 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
87ralbidv 3127 . . . . . . . . . . 11 (𝑦 = 𝑌 → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
94, 8rspc2v 3552 . . . . . . . . . 10 ((𝑋𝐴𝑌𝐴) → (∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) → ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
10 breq2 5037 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑋 <s 𝑧𝑋 <s 𝑤))
11 breq1 5036 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑧 <s 𝑌𝑤 <s 𝑌))
1210, 11anbi12d 634 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑋 <s 𝑧𝑧 <s 𝑌) ↔ (𝑋 <s 𝑤𝑤 <s 𝑌)))
13 eleq1w 2835 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
1412, 13imbi12d 349 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) ↔ ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
1514rspcv 3537 . . . . . . . . . . . . 13 (𝑤 No → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) → ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
16 bdaydm 33535 . . . . . . . . . . . . . . . . . . . . . 22 dom bday = No
1716sseq2i 3922 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday 𝐴 No )
18 bdayfun 33533 . . . . . . . . . . . . . . . . . . . . . 22 Fun bday
19 funfvima2 6986 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun bday 𝐴 ⊆ dom bday ) → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2018, 19mpan 690 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2117, 20sylbir 238 . . . . . . . . . . . . . . . . . . . 20 (𝐴 No → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2221imp 411 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝑤) ∈ ( bday 𝐴))
23 intss1 4854 . . . . . . . . . . . . . . . . . . 19 (( bday 𝑤) ∈ ( bday 𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
2422, 23syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
25 imassrn 5913 . . . . . . . . . . . . . . . . . . . . 21 ( bday 𝐴) ⊆ ran bday
26 bdayrn 33536 . . . . . . . . . . . . . . . . . . . . 21 ran bday = On
2725, 26sseqtri 3929 . . . . . . . . . . . . . . . . . . . 20 ( bday 𝐴) ⊆ On
2822ne0d 4235 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ≠ ∅)
29 oninton 7515 . . . . . . . . . . . . . . . . . . . 20 ((( bday 𝐴) ⊆ On ∧ ( bday 𝐴) ≠ ∅) → ( bday 𝐴) ∈ On)
3027, 28, 29sylancr 591 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ∈ On)
31 bdayelon 33537 . . . . . . . . . . . . . . . . . . 19 ( bday 𝑤) ∈ On
32 ontri1 6204 . . . . . . . . . . . . . . . . . . 19 (( ( bday 𝐴) ∈ On ∧ ( bday 𝑤) ∈ On) → ( ( bday 𝐴) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3330, 31, 32sylancl 590 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( ( bday 𝐴) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3424, 33mpbid 235 . . . . . . . . . . . . . . . . 17 ((𝐴 No 𝑤𝐴) → ¬ ( bday 𝑤) ∈ ( bday 𝐴))
3534ex 417 . . . . . . . . . . . . . . . 16 (𝐴 No → (𝑤𝐴 → ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
36 eleq2 2841 . . . . . . . . . . . . . . . . . 18 (( bday 𝑋) = ( bday 𝐴) → (( bday 𝑤) ∈ ( bday 𝑋) ↔ ( bday 𝑤) ∈ ( bday 𝐴)))
3736notbid 322 . . . . . . . . . . . . . . . . 17 (( bday 𝑋) = ( bday 𝐴) → (¬ ( bday 𝑤) ∈ ( bday 𝑋) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3837biimprcd 253 . . . . . . . . . . . . . . . 16 (¬ ( bday 𝑤) ∈ ( bday 𝐴) → (( bday 𝑋) = ( bday 𝐴) → ¬ ( bday 𝑤) ∈ ( bday 𝑋)))
3935, 38syl6 35 . . . . . . . . . . . . . . 15 (𝐴 No → (𝑤𝐴 → (( bday 𝑋) = ( bday 𝐴) → ¬ ( bday 𝑤) ∈ ( bday 𝑋))))
4039com3l 89 . . . . . . . . . . . . . 14 (𝑤𝐴 → (( bday 𝑋) = ( bday 𝐴) → (𝐴 No → ¬ ( bday 𝑤) ∈ ( bday 𝑋))))
4140adantrd 496 . . . . . . . . . . . . 13 (𝑤𝐴 → ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → (𝐴 No → ¬ ( bday 𝑤) ∈ ( bday 𝑋))))
4215, 41syl8 76 . . . . . . . . . . . 12 (𝑤 No → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) → ((𝑋 <s 𝑤𝑤 <s 𝑌) → ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → (𝐴 No → ¬ ( bday 𝑤) ∈ ( bday 𝑋))))))
4342com35 98 . . . . . . . . . . 11 (𝑤 No → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) → (𝐴 No → ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → ((𝑋 <s 𝑤𝑤 <s 𝑌) → ¬ ( bday 𝑤) ∈ ( bday 𝑋))))))
4443com4l 92 . . . . . . . . . 10 (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) → (𝐴 No → ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → (𝑤 No → ((𝑋 <s 𝑤𝑤 <s 𝑌) → ¬ ( bday 𝑤) ∈ ( bday 𝑋))))))
459, 44syl6 35 . . . . . . . . 9 ((𝑋𝐴𝑌𝐴) → (∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) → (𝐴 No → ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → (𝑤 No → ((𝑋 <s 𝑤𝑤 <s 𝑌) → ¬ ( bday 𝑤) ∈ ( bday 𝑋)))))))
4645com3l 89 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) → (𝐴 No → ((𝑋𝐴𝑌𝐴) → ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → (𝑤 No → ((𝑋 <s 𝑤𝑤 <s 𝑌) → ¬ ( bday 𝑤) ∈ ( bday 𝑋)))))))
4746impcom 412 . . . . . . 7 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ((𝑋𝐴𝑌𝐴) → ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → (𝑤 No → ((𝑋 <s 𝑤𝑤 <s 𝑌) → ¬ ( bday 𝑤) ∈ ( bday 𝑋))))))
4847imp42 431 . . . . . 6 ((((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) ∧ 𝑤 No ) → ((𝑋 <s 𝑤𝑤 <s 𝑌) → ¬ ( bday 𝑤) ∈ ( bday 𝑋)))
4948con2d 136 . . . . 5 ((((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) ∧ 𝑤 No ) → (( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)))
50 3anass 1093 . . . . . . 7 ((( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5150notbii 324 . . . . . 6 (¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
52 imnan 404 . . . . . 6 ((( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)) ↔ ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5351, 52bitr4i 281 . . . . 5 (¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ (( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5449, 53sylibr 237 . . . 4 ((((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) ∧ 𝑤 No ) → ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
5554nrexdv 3195 . . 3 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ¬ ∃𝑤 No (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
56 ssel 3886 . . . . . . . . 9 (𝐴 No → (𝑋𝐴𝑋 No ))
57 ssel 3886 . . . . . . . . 9 (𝐴 No → (𝑌𝐴𝑌 No ))
5856, 57anim12d 612 . . . . . . . 8 (𝐴 No → ((𝑋𝐴𝑌𝐴) → (𝑋 No 𝑌 No )))
5958imp 411 . . . . . . 7 ((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) → (𝑋 No 𝑌 No ))
60 eqtr3 2781 . . . . . . 7 ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → ( bday 𝑋) = ( bday 𝑌))
6159, 60anim12i 616 . . . . . 6 (((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6261anasss 471 . . . . 5 ((𝐴 No ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6362adantlr 715 . . . 4 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
64 nodense 33461 . . . . 5 (((𝑋 No 𝑌 No ) ∧ (( bday 𝑋) = ( bday 𝑌) ∧ 𝑋 <s 𝑌)) → ∃𝑤 No (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
6564anassrs 472 . . . 4 ((((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ∃𝑤 No (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
6663, 65sylan 584 . . 3 ((((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) ∧ 𝑋 <s 𝑌) → ∃𝑤 No (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
6755, 66mtand 816 . 2 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ¬ 𝑋 <s 𝑌)
6867ex 417 1 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ¬ 𝑋 <s 𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  ∃wrex 3072   ⊆ wss 3859  ∅c0 4226  ∩ cint 4839   class class class wbr 5033  dom cdm 5525  ran crn 5526   “ cima 5528  Oncon0 6170  Fun wfun 6330  ‘cfv 6336   No csur 33409
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