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Theorem nocvxminlem 33131
Description: Lemma for nocvxmin 33132. 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 5066 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 <s 𝑧𝑋 <s 𝑧))
21anbi1d 629 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑦)))
32imbi1d 343 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
43ralbidv 3202 . . . . . . . . . . 11 (𝑥 = 𝑋 → (∀𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)))
5 breq2 5067 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑧 <s 𝑦𝑧 <s 𝑌))
65anbi2d 628 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑋 <s 𝑧𝑧 <s 𝑦) ↔ (𝑋 <s 𝑧𝑧 <s 𝑌)))
76imbi1d 343 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
87ralbidv 3202 . . . . . . . . . . 11 (𝑦 = 𝑌 → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) ↔ ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
94, 8rspc2v 3637 . . . . . . . . . 10 ((𝑋𝐴𝑌𝐴) → (∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴) → ∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴)))
10 breq2 5067 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑋 <s 𝑧𝑋 <s 𝑤))
11 breq1 5066 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑧 <s 𝑌𝑤 <s 𝑌))
1210, 11anbi12d 630 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑋 <s 𝑧𝑧 <s 𝑌) ↔ (𝑋 <s 𝑤𝑤 <s 𝑌)))
13 eleq1w 2900 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
1412, 13imbi12d 346 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) ↔ ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
1514rspcv 3622 . . . . . . . . . . . . 13 (𝑤 No → (∀𝑧 No ((𝑋 <s 𝑧𝑧 <s 𝑌) → 𝑧𝐴) → ((𝑋 <s 𝑤𝑤 <s 𝑌) → 𝑤𝐴)))
16 bdaydm 33128 . . . . . . . . . . . . . . . . . . . . . 22 dom bday = No
1716sseq2i 4000 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday 𝐴 No )
18 bdayfun 33126 . . . . . . . . . . . . . . . . . . . . . 22 Fun bday
19 funfvima2 6988 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun bday 𝐴 ⊆ dom bday ) → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2018, 19mpan 686 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ dom bday → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2117, 20sylbir 236 . . . . . . . . . . . . . . . . . . . 20 (𝐴 No → (𝑤𝐴 → ( bday 𝑤) ∈ ( bday 𝐴)))
2221imp 407 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝑤) ∈ ( bday 𝐴))
23 intss1 4889 . . . . . . . . . . . . . . . . . . 19 (( bday 𝑤) ∈ ( bday 𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
2422, 23syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ⊆ ( bday 𝑤))
25 imassrn 5938 . . . . . . . . . . . . . . . . . . . . 21 ( bday 𝐴) ⊆ ran bday
26 bdayrn 33129 . . . . . . . . . . . . . . . . . . . . 21 ran bday = On
2725, 26sseqtri 4007 . . . . . . . . . . . . . . . . . . . 20 ( bday 𝐴) ⊆ On
2822ne0d 4305 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ≠ ∅)
29 oninton 7503 . . . . . . . . . . . . . . . . . . . 20 ((( bday 𝐴) ⊆ On ∧ ( bday 𝐴) ≠ ∅) → ( bday 𝐴) ∈ On)
3027, 28, 29sylancr 587 . . . . . . . . . . . . . . . . . . 19 ((𝐴 No 𝑤𝐴) → ( bday 𝐴) ∈ On)
31 bdayelon 33130 . . . . . . . . . . . . . . . . . . 19 ( bday 𝑤) ∈ On
32 ontri1 6223 . . . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . . 17 ((𝐴 No 𝑤𝐴) → ¬ ( bday 𝑤) ∈ ( bday 𝐴))
3534ex 413 . . . . . . . . . . . . . . . 16 (𝐴 No → (𝑤𝐴 → ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
36 eleq2 2906 . . . . . . . . . . . . . . . . . 18 (( bday 𝑋) = ( bday 𝐴) → (( bday 𝑤) ∈ ( bday 𝑋) ↔ ( bday 𝑤) ∈ ( bday 𝐴)))
3736notbid 319 . . . . . . . . . . . . . . . . 17 (( bday 𝑋) = ( bday 𝐴) → (¬ ( bday 𝑤) ∈ ( bday 𝑋) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝐴)))
3837biimprcd 251 . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . 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 408 . . . . . . 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 427 . . . . . 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 1089 . . . . . . 7 ((( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5150notbii 321 . . . . . 6 (¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
52 imnan 400 . . . . . 6 ((( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)) ↔ ¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5351, 52bitr4i 279 . . . . 5 (¬ (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌) ↔ (( bday 𝑤) ∈ ( bday 𝑋) → ¬ (𝑋 <s 𝑤𝑤 <s 𝑌)))
5449, 53sylibr 235 . . . 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 3275 . . 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 3965 . . . . . . . . 9 (𝐴 No → (𝑋𝐴𝑋 No ))
57 ssel 3965 . . . . . . . . 9 (𝐴 No → (𝑌𝐴𝑌 No ))
5856, 57anim12d 608 . . . . . . . 8 (𝐴 No → ((𝑋𝐴𝑌𝐴) → (𝑋 No 𝑌 No )))
5958imp 407 . . . . . . 7 ((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) → (𝑋 No 𝑌 No ))
60 eqtr3 2848 . . . . . . 7 ((( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)) → ( bday 𝑋) = ( bday 𝑌))
6159, 60anim12i 612 . . . . . 6 (((𝐴 No ∧ (𝑋𝐴𝑌𝐴)) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6261anasss 467 . . . . 5 ((𝐴 No ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ((𝑋 No 𝑌 No ) ∧ ( bday 𝑋) = ( bday 𝑌)))
6362adantlr 711 . . . 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 33080 . . . . 5 (((𝑋 No 𝑌 No ) ∧ (( bday 𝑋) = ( bday 𝑌) ∧ 𝑋 <s 𝑌)) → ∃𝑤 No (( bday 𝑤) ∈ ( bday 𝑋) ∧ 𝑋 <s 𝑤𝑤 <s 𝑌))
6564anassrs 468 . . . 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 812 . 2 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴)))) → ¬ 𝑋 <s 𝑌)
6867ex 413 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  wss 3940  c0 4295   cint 4874   class class class wbr 5063  dom cdm 5554  ran crn 5555  cima 5557  Oncon0 6189  Fun wfun 6346  cfv 6352   No csur 33031   <s cslt 33032   bday cbday 33033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-1o 8093  df-2o 8094  df-no 33034  df-slt 33035  df-bday 33036
This theorem is referenced by:  nocvxmin  33132
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