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Theorem nocvxminlem 32212
Description: Lemma for nocvxmin 32213. 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 4847 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 <s 𝑧𝑋 <s 𝑧))
21anbi1d 617 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑦)))
32imbi1d 332 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
43ralbidv 3174 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
5 breq2 4848 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑧 <s 𝑦𝑧 <s 𝑌))
65anbi2d 616 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑋 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑌)))
76imbi1d 332 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
87ralbidv 3174 . . . . . . . . . . 11 (𝑦 = 𝑌 → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
94, 8rspc2v 3515 . . . . . . . . . 10 ((𝑋𝐴𝑌𝐴) → (∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) → ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
10 breq2 4848 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑋 <s 𝑧𝑋 <s 𝑤))
11 breq1 4847 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑧 <s 𝑌𝑤 <s 𝑌))
1210, 11anbi12d 618 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑋 <s 𝑧𝑧 <s 𝑌) ↔ (𝑋 <s 𝑤𝑤 <s 𝑌)))
13 eleq1w 2868 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
1412, 13imbi12d 335 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) ↔ ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
1514rspcv 3498 . . . . . . . . . . . . 13 (𝑤 No → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) → ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
16 bdaydm 32209 . . . . . . . . . . . . . . . . . . . . . 22 dom bday = No
1716sseq2i 3827 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday 𝐴 No )
18 bdayfun 32207 . . . . . . . . . . . . . . . . . . . . . 22 Fun bday
19 funfvima2 6714 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun bday 𝐴 ⊆ dom bday ) → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2018, 19mpan 673 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2117, 20sylbir 226 . . . . . . . . . . . . . . . . . . . 20 (𝐴 No → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2221imp 395 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝑤) ∈ ( bday 𝐴))
23 intss1 4684 . . . . . . . . . . . . . . . . . . 19 (( bday 𝑤) ∈ ( bday 𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
2422, 23syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
25 imassrn 5687 . . . . . . . . . . . . . . . . . . . . 21 ( bday 𝐴) ⊆ ran bday
26 bdayrn 32210 . . . . . . . . . . . . . . . . . . . . 21 ran bday = On
2725, 26sseqtri 3834 . . . . . . . . . . . . . . . . . . . 20 ( bday 𝐴) ⊆ On
2822ne0d 4123 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ≠ ∅)
29 oninton 7226 . . . . . . . . . . . . . . . . . . . 20 ((( bday 𝐴) ⊆ On ∧ ( bday 𝐴) ≠ ∅) → ( bday 𝐴) ∈ On)
3027, 28, 29sylancr 577 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ∈ On)
31 bdayelon 32211 . . . . . . . . . . . . . . . . . . 19 ( bday 𝑤) ∈ On
32 ontri1 5970 . . . . . . . . . . . . . . . . . . 19 (( ( bday 𝐴) ∈ On ∧ ( bday 𝑤) ∈ On) → ( ( bday 𝐴) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3330, 31, 32sylancl 576 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( ( bday 𝐴) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3424, 33mpbid 223 . . . . . . . . . . . . . . . . 17 ((𝐴 No 𝑤𝐴) → ¬ ( bday 𝑤) ∈ ( bday 𝐴))
3534ex 399 . . . . . . . . . . . . . . . 16 (𝐴 No → (𝑤𝐴 → ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
36 eleq2 2874 . . . . . . . . . . . . . . . . . 18 (( bday 𝑋) = ( bday 𝐴) → (( bday 𝑤) ∈ ( bday 𝑋) ↔ ( bday 𝑤) ∈ ( bday 𝐴)))
3736notbid 309 . . . . . . . . . . . . . . . . 17 (( bday 𝑋) = ( bday 𝐴) → (¬ ( bday 𝑤) ∈ ( bday 𝑋) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3837biimprcd 241 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 396 . . . . . . 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 415 . . . . . 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 131 . . . . 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 1109 . . . . . . 7 ((( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5150notbii 311 . . . . . 6 (¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
52 imnan 388 . . . . . 6 ((( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)) ↔ ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5351, 52bitr4i 269 . . . . 5 (¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ (( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5449, 53sylibr 225 . . . 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 3188 . . 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 3792 . . . . . . . . 9 (𝐴 No → (𝑋𝐴𝑋 No ))
57 ssel 3792 . . . . . . . . 9 (𝐴 No → (𝑌𝐴𝑌 No ))
5856, 57anim12d 598 . . . . . . . 8 (𝐴 No → ((𝑋𝐴𝑌𝐴) → (𝑋 No 𝑌 No )))
5958imp 395 . . . . . . 7 ((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) → (𝑋 No 𝑌 No ))
60 eqtr3 2827 . . . . . . 7 ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → ( bday 𝑋) = ( bday 𝑌))
6159, 60anim12i 602 . . . . . 6 (((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6261anasss 454 . . . . 5 ((𝐴 No ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6362adantlr 697 . . . 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 32161 . . . . 5 (((𝑋 No 𝑌 No ) ∧ (( bday 𝑋) = ( bday 𝑌) ∧ 𝑋 <s 𝑌)) → ∃𝑤 No (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
6564anassrs 455 . . . 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 571 . . 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 841 . 2 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ¬ 𝑋 <s 𝑌)
6867ex 399 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wral 3096  wrex 3097  wss 3769  c0 4116   cint 4669   class class class wbr 4844  dom cdm 5311  ran crn 5312  cima 5314  Oncon0 5936  Fun wfun 6091  cfv 6097   No csur 32112   <s cslt 32113   bday cbday 32114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-1o 7792  df-2o 7793  df-no 32115  df-slt 32116  df-bday 32117
This theorem is referenced by:  nocvxmin  32213
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