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Theorem negsprop 34321
Description: Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
Assertion
Ref Expression
negsprop ((𝐴 No 𝐵 No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))

Proof of Theorem negsprop
Dummy variables 𝑎 𝑏 𝑝 𝑞 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdayelon 27067 . . . 4 ( bday 𝐴) ∈ On
2 bdayelon 27067 . . . 4 ( bday 𝐵) ∈ On
31, 2onun2i 6436 . . 3 (( bday 𝐴) ∪ ( bday 𝐵)) ∈ On
4 risset 3219 . . 3 ((( bday 𝐴) ∪ ( bday 𝐵)) ∈ On ↔ ∃𝑎 ∈ On 𝑎 = (( bday 𝐴) ∪ ( bday 𝐵)))
53, 4mpbi 229 . 2 𝑎 ∈ On 𝑎 = (( bday 𝐴) ∪ ( bday 𝐵))
6 eqeq1 2741 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) ↔ 𝑏 = (( bday 𝑝) ∪ ( bday 𝑞))))
76imbi1d 341 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ (𝑏 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
872ralbidv 3210 . . . . . . 7 (𝑎 = 𝑏 → (∀𝑝 No 𝑞 No (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ ∀𝑝 No 𝑞 No (𝑏 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
9 fveq2 6839 . . . . . . . . . . 11 (𝑝 = 𝑥 → ( bday 𝑝) = ( bday 𝑥))
109uneq1d 4120 . . . . . . . . . 10 (𝑝 = 𝑥 → (( bday 𝑝) ∪ ( bday 𝑞)) = (( bday 𝑥) ∪ ( bday 𝑞)))
1110eqeq2d 2748 . . . . . . . . 9 (𝑝 = 𝑥 → (𝑏 = (( bday 𝑝) ∪ ( bday 𝑞)) ↔ 𝑏 = (( bday 𝑥) ∪ ( bday 𝑞))))
12 fveq2 6839 . . . . . . . . . . 11 (𝑝 = 𝑥 → ( -us ‘𝑝) = ( -us ‘𝑥))
1312eleq1d 2822 . . . . . . . . . 10 (𝑝 = 𝑥 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝑥) ∈ No ))
14 breq1 5106 . . . . . . . . . . 11 (𝑝 = 𝑥 → (𝑝 <s 𝑞𝑥 <s 𝑞))
1512breq2d 5115 . . . . . . . . . . 11 (𝑝 = 𝑥 → (( -us ‘𝑞) <s ( -us ‘𝑝) ↔ ( -us ‘𝑞) <s ( -us ‘𝑥)))
1614, 15imbi12d 344 . . . . . . . . . 10 (𝑝 = 𝑥 → ((𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)) ↔ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥))))
1713, 16anbi12d 631 . . . . . . . . 9 (𝑝 = 𝑥 → ((( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))) ↔ (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥)))))
1811, 17imbi12d 344 . . . . . . . 8 (𝑝 = 𝑥 → ((𝑏 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ (𝑏 = (( bday 𝑥) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥))))))
19 fveq2 6839 . . . . . . . . . . 11 (𝑞 = 𝑦 → ( bday 𝑞) = ( bday 𝑦))
2019uneq2d 4121 . . . . . . . . . 10 (𝑞 = 𝑦 → (( bday 𝑥) ∪ ( bday 𝑞)) = (( bday 𝑥) ∪ ( bday 𝑦)))
2120eqeq2d 2748 . . . . . . . . 9 (𝑞 = 𝑦 → (𝑏 = (( bday 𝑥) ∪ ( bday 𝑞)) ↔ 𝑏 = (( bday 𝑥) ∪ ( bday 𝑦))))
22 breq2 5107 . . . . . . . . . . 11 (𝑞 = 𝑦 → (𝑥 <s 𝑞𝑥 <s 𝑦))
23 fveq2 6839 . . . . . . . . . . . 12 (𝑞 = 𝑦 → ( -us ‘𝑞) = ( -us ‘𝑦))
2423breq1d 5113 . . . . . . . . . . 11 (𝑞 = 𝑦 → (( -us ‘𝑞) <s ( -us ‘𝑥) ↔ ( -us ‘𝑦) <s ( -us ‘𝑥)))
2522, 24imbi12d 344 . . . . . . . . . 10 (𝑞 = 𝑦 → ((𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥)) ↔ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))
2625anbi2d 629 . . . . . . . . 9 (𝑞 = 𝑦 → ((( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥))) ↔ (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
2721, 26imbi12d 344 . . . . . . . 8 (𝑞 = 𝑦 → ((𝑏 = (( bday 𝑥) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑥)))) ↔ (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))))
2818, 27cbvral2vw 3225 . . . . . . 7 (∀𝑝 No 𝑞 No (𝑏 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ ∀𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
298, 28bitrdi 286 . . . . . 6 (𝑎 = 𝑏 → (∀𝑝 No 𝑞 No (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ ∀𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))))
30 raleq 3307 . . . . . . . . . . 11 (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (∀𝑏𝑎𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))∀𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))))
31 ralrot3 3274 . . . . . . . . . . . 12 (∀𝑥 No 𝑦 No 𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))(𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))∀𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
32 r19.23v 3177 . . . . . . . . . . . . . 14 (∀𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))(𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ (∃𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
33 risset 3219 . . . . . . . . . . . . . . 15 ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) ↔ ∃𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)))
3433imbi1i 349 . . . . . . . . . . . . . 14 (((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ (∃𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
3532, 34bitr4i 277 . . . . . . . . . . . . 13 (∀𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))(𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
36352ralbii 3125 . . . . . . . . . . . 12 (∀𝑥 No 𝑦 No 𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))(𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
3731, 36bitr3i 276 . . . . . . . . . . 11 (∀𝑏 ∈ (( bday 𝑝) ∪ ( bday 𝑞))∀𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
3830, 37bitrdi 286 . . . . . . . . . 10 (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (∀𝑏𝑎𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))))
39 simpr 485 . . . . . . . . . . . . . 14 (((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
40 simpll 765 . . . . . . . . . . . . . 14 (((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → 𝑝 No )
4139, 40negsproplem3 34316 . . . . . . . . . . . . 13 (((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → (( -us ‘𝑝) ∈ No ∧ ( -us “ ( R ‘𝑝)) <<s {( -us ‘𝑝)} ∧ {( -us ‘𝑝)} <<s ( -us “ ( L ‘𝑝))))
4241simp1d 1142 . . . . . . . . . . . 12 (((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → ( -us ‘𝑝) ∈ No )
43 simplr 767 . . . . . . . . . . . . . 14 ((((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
44 simplll 773 . . . . . . . . . . . . . 14 ((((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑝 No )
45 simpllr 774 . . . . . . . . . . . . . 14 ((((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑞 No )
46 simpr 485 . . . . . . . . . . . . . 14 ((((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → 𝑝 <s 𝑞)
4743, 44, 45, 46negsproplem7 34320 . . . . . . . . . . . . 13 ((((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) ∧ 𝑝 <s 𝑞) → ( -us ‘𝑞) <s ( -us ‘𝑝))
4847ex 413 . . . . . . . . . . . 12 (((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))
4942, 48jca 512 . . . . . . . . . . 11 (((𝑝 No 𝑞 No ) ∧ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))
5049expcom 414 . . . . . . . . . 10 (∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ((𝑝 No 𝑞 No ) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))))
5138, 50syl6bi 252 . . . . . . . . 9 (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (∀𝑏𝑎𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ((𝑝 No 𝑞 No ) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
5251com3l 89 . . . . . . . 8 (∀𝑏𝑎𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ((𝑝 No 𝑞 No ) → (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
5352ralrimivv 3193 . . . . . . 7 (∀𝑏𝑎𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ∀𝑝 No 𝑞 No (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))))
5453a1i 11 . . . . . 6 (𝑎 ∈ On → (∀𝑏𝑎𝑥 No 𝑦 No (𝑏 = (( bday 𝑥) ∪ ( bday 𝑦)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ∀𝑝 No 𝑞 No (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))))))
5529, 54tfis2 7785 . . . . 5 (𝑎 ∈ On → ∀𝑝 No 𝑞 No (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))))
56 fveq2 6839 . . . . . . . . 9 (𝑝 = 𝐴 → ( bday 𝑝) = ( bday 𝐴))
5756uneq1d 4120 . . . . . . . 8 (𝑝 = 𝐴 → (( bday 𝑝) ∪ ( bday 𝑞)) = (( bday 𝐴) ∪ ( bday 𝑞)))
5857eqeq2d 2748 . . . . . . 7 (𝑝 = 𝐴 → (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) ↔ 𝑎 = (( bday 𝐴) ∪ ( bday 𝑞))))
59 fveq2 6839 . . . . . . . . 9 (𝑝 = 𝐴 → ( -us ‘𝑝) = ( -us ‘𝐴))
6059eleq1d 2822 . . . . . . . 8 (𝑝 = 𝐴 → (( -us ‘𝑝) ∈ No ↔ ( -us ‘𝐴) ∈ No ))
61 breq1 5106 . . . . . . . . 9 (𝑝 = 𝐴 → (𝑝 <s 𝑞𝐴 <s 𝑞))
6259breq2d 5115 . . . . . . . . 9 (𝑝 = 𝐴 → (( -us ‘𝑞) <s ( -us ‘𝑝) ↔ ( -us ‘𝑞) <s ( -us ‘𝐴)))
6361, 62imbi12d 344 . . . . . . . 8 (𝑝 = 𝐴 → ((𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)) ↔ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴))))
6460, 63anbi12d 631 . . . . . . 7 (𝑝 = 𝐴 → ((( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝))) ↔ (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴)))))
6558, 64imbi12d 344 . . . . . 6 (𝑝 = 𝐴 → ((𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) ↔ (𝑎 = (( bday 𝐴) ∪ ( bday 𝑞)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴))))))
66 fveq2 6839 . . . . . . . . 9 (𝑞 = 𝐵 → ( bday 𝑞) = ( bday 𝐵))
6766uneq2d 4121 . . . . . . . 8 (𝑞 = 𝐵 → (( bday 𝐴) ∪ ( bday 𝑞)) = (( bday 𝐴) ∪ ( bday 𝐵)))
6867eqeq2d 2748 . . . . . . 7 (𝑞 = 𝐵 → (𝑎 = (( bday 𝐴) ∪ ( bday 𝑞)) ↔ 𝑎 = (( bday 𝐴) ∪ ( bday 𝐵))))
69 breq2 5107 . . . . . . . . 9 (𝑞 = 𝐵 → (𝐴 <s 𝑞𝐴 <s 𝐵))
70 fveq2 6839 . . . . . . . . . 10 (𝑞 = 𝐵 → ( -us ‘𝑞) = ( -us ‘𝐵))
7170breq1d 5113 . . . . . . . . 9 (𝑞 = 𝐵 → (( -us ‘𝑞) <s ( -us ‘𝐴) ↔ ( -us ‘𝐵) <s ( -us ‘𝐴)))
7269, 71imbi12d 344 . . . . . . . 8 (𝑞 = 𝐵 → ((𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴)) ↔ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))
7372anbi2d 629 . . . . . . 7 (𝑞 = 𝐵 → ((( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴))) ↔ (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴)))))
7468, 73imbi12d 344 . . . . . 6 (𝑞 = 𝐵 → ((𝑎 = (( bday 𝐴) ∪ ( bday 𝑞)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝐴)))) ↔ (𝑎 = (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))))
7565, 74rspc2v 3588 . . . . 5 ((𝐴 No 𝐵 No ) → (∀𝑝 No 𝑞 No (𝑎 = (( bday 𝑝) ∪ ( bday 𝑞)) → (( -us ‘𝑝) ∈ No ∧ (𝑝 <s 𝑞 → ( -us ‘𝑞) <s ( -us ‘𝑝)))) → (𝑎 = (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))))
7655, 75syl5com 31 . . . 4 (𝑎 ∈ On → ((𝐴 No 𝐵 No ) → (𝑎 = (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))))
7776com23 86 . . 3 (𝑎 ∈ On → (𝑎 = (( bday 𝐴) ∪ ( bday 𝐵)) → ((𝐴 No 𝐵 No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))))
7877rexlimiv 3143 . 2 (∃𝑎 ∈ On 𝑎 = (( bday 𝐴) ∪ ( bday 𝐵)) → ((𝐴 No 𝐵 No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴)))))
795, 78ax-mp 5 1 ((𝐴 No 𝐵 No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062  wrex 3071  cun 3906  {csn 4584   class class class wbr 5103  cima 5634  Oncon0 6315  cfv 6493   No csur 26939   <s cslt 26940   bday cbday 26941   <<s csslt 27071   L cleft 27126   R cright 27127   -us cnegs 34306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-1o 8404  df-2o 8405  df-no 26942  df-slt 26943  df-bday 26944  df-sslt 27072  df-scut 27074  df-0s 27114  df-made 27128  df-old 27129  df-left 27131  df-right 27132  df-norec 34246  df-negs 34308
This theorem is referenced by:  negscl  34322  sltnegim  34324  negscut  34325
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