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Theorem nocvxminlem 27822
Description: Lemma for nocvxmin 27823. 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 5146 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 <s 𝑧𝑋 <s 𝑧))
21anbi1d 631 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑦)))
32imbi1d 341 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
43ralbidv 3178 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
5 breq2 5147 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑧 <s 𝑦𝑧 <s 𝑌))
65anbi2d 630 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑋 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑌)))
76imbi1d 341 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
87ralbidv 3178 . . . . . . . . . . 11 (𝑦 = 𝑌 → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
94, 8rspc2v 3633 . . . . . . . . . 10 ((𝑋𝐴𝑌𝐴) → (∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) → ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
10 breq2 5147 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑋 <s 𝑧𝑋 <s 𝑤))
11 breq1 5146 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑧 <s 𝑌𝑤 <s 𝑌))
1210, 11anbi12d 632 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑋 <s 𝑧𝑧 <s 𝑌) ↔ (𝑋 <s 𝑤𝑤 <s 𝑌)))
13 eleq1w 2824 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
1412, 13imbi12d 344 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) ↔ ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
1514rspcv 3618 . . . . . . . . . . . . 13 (𝑤 No → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) → ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
16 bdaydm 27819 . . . . . . . . . . . . . . . . . . . . . 22 dom bday = No
1716sseq2i 4013 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday 𝐴 No )
18 bdayfun 27817 . . . . . . . . . . . . . . . . . . . . . 22 Fun bday
19 funfvima2 7251 . . . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . . . 20 (𝐴 No → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2221imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝑤) ∈ ( bday 𝐴))
23 intss1 4963 . . . . . . . . . . . . . . . . . . 19 (( bday 𝑤) ∈ ( bday 𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
2422, 23syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
25 imassrn 6089 . . . . . . . . . . . . . . . . . . . . 21 ( bday 𝐴) ⊆ ran bday
26 bdayrn 27820 . . . . . . . . . . . . . . . . . . . . 21 ran bday = On
2725, 26sseqtri 4032 . . . . . . . . . . . . . . . . . . . 20 ( bday 𝐴) ⊆ On
2822ne0d 4342 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ≠ ∅)
29 oninton 7815 . . . . . . . . . . . . . . . . . . . 20 ((( bday 𝐴) ⊆ On ∧ ( bday 𝐴) ≠ ∅) → ( bday 𝐴) ∈ On)
3027, 28, 29sylancr 587 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ∈ On)
31 bdayelon 27821 . . . . . . . . . . . . . . . . . . 19 ( bday 𝑤) ∈ On
32 ontri1 6418 . . . . . . . . . . . . . . . . . . 19 (( ( bday 𝐴) ∈ On ∧ ( bday 𝑤) ∈ On) → ( ( bday 𝐴) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3330, 31, 32sylancl 586 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( ( bday 𝐴) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3424, 33mpbid 232 . . . . . . . . . . . . . . . . 17 ((𝐴 No 𝑤𝐴) → ¬ ( bday 𝑤) ∈ ( bday 𝐴))
3534ex 412 . . . . . . . . . . . . . . . 16 (𝐴 No → (𝑤𝐴 → ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
36 eleq2 2830 . . . . . . . . . . . . . . . . . 18 (( bday 𝑋) = ( bday 𝐴) → (( bday 𝑤) ∈ ( bday 𝑋) ↔ ( bday 𝑤) ∈ ( bday 𝐴)))
3736notbid 318 . . . . . . . . . . . . . . . . 17 (( bday 𝑋) = ( bday 𝐴) → (¬ ( bday 𝑤) ∈ ( bday 𝑋) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3837biimprcd 250 . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . 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 407 . . . . . . 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 426 . . . . . 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 134 . . . . 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 1095 . . . . . . 7 ((( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5150notbii 320 . . . . . 6 (¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
52 imnan 399 . . . . . 6 ((( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)) ↔ ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5351, 52bitr4i 278 . . . . 5 (¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ (( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5449, 53sylibr 234 . . . 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 3149 . . 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 3977 . . . . . . . . 9 (𝐴 No → (𝑋𝐴𝑋 No ))
57 ssel 3977 . . . . . . . . 9 (𝐴 No → (𝑌𝐴𝑌 No ))
5856, 57anim12d 609 . . . . . . . 8 (𝐴 No → ((𝑋𝐴𝑌𝐴) → (𝑋 No 𝑌 No )))
5958imp 406 . . . . . . 7 ((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) → (𝑋 No 𝑌 No ))
60 eqtr3 2763 . . . . . . 7 ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → ( bday 𝑋) = ( bday 𝑌))
6159, 60anim12i 613 . . . . . 6 (((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6261anasss 466 . . . . 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 27737 . . . . 5 (((𝑋 No 𝑌 No ) ∧ (( bday 𝑋) = ( bday 𝑌) ∧ 𝑋 <s 𝑌)) → ∃𝑤 No (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
6564anassrs 467 . . . 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 580 . . 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 412 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333   cint 4946   class class class wbr 5143  dom cdm 5685  ran crn 5686  cima 5688  Oncon0 6384  Fun wfun 6555  cfv 6561   No csur 27684   <s cslt 27685   bday cbday 27686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689
This theorem is referenced by:  nocvxmin  27823
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