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Theorem nocvxminlem 27139
Description: Lemma for nocvxmin 27140. 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 5109 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 <s 𝑧𝑋 <s 𝑧))
21anbi1d 631 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑦)))
32imbi1d 342 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
43ralbidv 3171 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
5 breq2 5110 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑧 <s 𝑦𝑧 <s 𝑌))
65anbi2d 630 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑋 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑌)))
76imbi1d 342 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
87ralbidv 3171 . . . . . . . . . . 11 (𝑦 = 𝑌 → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
94, 8rspc2v 3589 . . . . . . . . . 10 ((𝑋𝐴𝑌𝐴) → (∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) → ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
10 breq2 5110 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑋 <s 𝑧𝑋 <s 𝑤))
11 breq1 5109 . . . . . . . . . . . . . . . 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 2817 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
1412, 13imbi12d 345 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) ↔ ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
1514rspcv 3576 . . . . . . . . . . . . 13 (𝑤 No → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) → ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
16 bdaydm 27136 . . . . . . . . . . . . . . . . . . . . . 22 dom bday = No
1716sseq2i 3974 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday 𝐴 No )
18 bdayfun 27134 . . . . . . . . . . . . . . . . . . . . . 22 Fun bday
19 funfvima2 7182 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun bday 𝐴 ⊆ dom bday ) → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2018, 19mpan 689 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2117, 20sylbir 234 . . . . . . . . . . . . . . . . . . . 20 (𝐴 No → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2221imp 408 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝑤) ∈ ( bday 𝐴))
23 intss1 4925 . . . . . . . . . . . . . . . . . . 19 (( bday 𝑤) ∈ ( bday 𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
2422, 23syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
25 imassrn 6025 . . . . . . . . . . . . . . . . . . . . 21 ( bday 𝐴) ⊆ ran bday
26 bdayrn 27137 . . . . . . . . . . . . . . . . . . . . 21 ran bday = On
2725, 26sseqtri 3981 . . . . . . . . . . . . . . . . . . . 20 ( bday 𝐴) ⊆ On
2822ne0d 4296 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ≠ ∅)
29 oninton 7731 . . . . . . . . . . . . . . . . . . . 20 ((( bday 𝐴) ⊆ On ∧ ( bday 𝐴) ≠ ∅) → ( bday 𝐴) ∈ On)
3027, 28, 29sylancr 588 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ∈ On)
31 bdayelon 27138 . . . . . . . . . . . . . . . . . . 19 ( bday 𝑤) ∈ On
32 ontri1 6352 . . . . . . . . . . . . . . . . . . 19 (( ( bday 𝐴) ∈ On ∧ ( bday 𝑤) ∈ On) → ( ( bday 𝐴) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3330, 31, 32sylancl 587 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( ( bday 𝐴) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3424, 33mpbid 231 . . . . . . . . . . . . . . . . 17 ((𝐴 No 𝑤𝐴) → ¬ ( bday 𝑤) ∈ ( bday 𝐴))
3534ex 414 . . . . . . . . . . . . . . . 16 (𝐴 No → (𝑤𝐴 → ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
36 eleq2 2823 . . . . . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . 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 409 . . . . . . 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 428 . . . . . 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 1096 . . . . . . 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 401 . . . . . 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 233 . . . 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 3143 . . 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 3938 . . . . . . . . 9 (𝐴 No → (𝑋𝐴𝑋 No ))
57 ssel 3938 . . . . . . . . 9 (𝐴 No → (𝑌𝐴𝑌 No ))
5856, 57anim12d 610 . . . . . . . 8 (𝐴 No → ((𝑋𝐴𝑌𝐴) → (𝑋 No 𝑌 No )))
5958imp 408 . . . . . . 7 ((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) → (𝑋 No 𝑌 No ))
60 eqtr3 2759 . . . . . . 7 ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → ( bday 𝑋) = ( bday 𝑌))
6159, 60anim12i 614 . . . . . 6 (((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6261anasss 468 . . . . 5 ((𝐴 No ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6362adantlr 714 . . . 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 27056 . . . . 5 (((𝑋 No 𝑌 No ) ∧ (( bday 𝑋) = ( bday 𝑌) ∧ 𝑋 <s 𝑌)) → ∃𝑤 No (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
6564anassrs 469 . . . 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 581 . . 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 815 . 2 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ¬ 𝑋 <s 𝑌)
6867ex 414 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  wss 3911  c0 4283   cint 4908   class class class wbr 5106  dom cdm 5634  ran crn 5635  cima 5637  Oncon0 6318  Fun wfun 6491  cfv 6497   No csur 27004   <s cslt 27005   bday cbday 27006
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 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009
This theorem is referenced by:  nocvxmin  27140
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