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Theorem ltexprlemdisj 7547
Description: Our constructed difference is disjoint. Lemma for ltexpri 7554. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
Assertion
Ref Expression
ltexprlemdisj (𝐴<P 𝐵 → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑞,𝐴   𝑥,𝐵,𝑦,𝑞   𝑥,𝐶,𝑦,𝑞

Proof of Theorem ltexprlemdisj
Dummy variables 𝑧 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsonq 7339 . . . . . 6 <Q Or Q
2 ltrelnq 7306 . . . . . 6 <Q ⊆ (Q × Q)
31, 2son2lpi 5000 . . . . 5 ¬ (𝑦 <Q 𝑧𝑧 <Q 𝑦)
4 ltrelpr 7446 . . . . . . . . . . . . . . . 16 <P ⊆ (P × P)
54brel 4656 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → (𝐴P𝐵P))
65simprd 113 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
7 prop 7416 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
86, 7syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 prltlu 7428 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
108, 9syl3an1 1261 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
11103expb 1194 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1211adantlr 469 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1312adantrll 476 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1413adantrrl 478 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
15 ltanqg 7341 . . . . . . . . . 10 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
1615adantl 275 . . . . . . . . 9 ((((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
175simpld 111 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐴P)
18 prop 7416 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1917, 18syl 14 . . . . . . . . . . . 12 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
20 elprnqu 7423 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2119, 20sylan 281 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2221ad2ant2r 501 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) → 𝑦Q)
2322adantrr 471 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦Q)
24 elprnql 7422 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
2519, 24sylan 281 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
2625ad2ant2r 501 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → 𝑧Q)
2726adantrl 470 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑧Q)
28 simplr 520 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑞Q)
29 addcomnqg 7322 . . . . . . . . . 10 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 275 . . . . . . . . 9 ((((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3116, 23, 27, 28, 30caovord2d 6011 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → (𝑦 <Q 𝑧 ↔ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞)))
3214, 31mpbird 166 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦 <Q 𝑧)
33 prltlu 7428 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑧 <Q 𝑦)
3419, 33syl3an1 1261 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑧 <Q 𝑦)
35343com23 1199 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧 <Q 𝑦)
36353expb 1194 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3736adantlr 469 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3837adantrlr 477 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3938adantrrr 479 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑧 <Q 𝑦)
4032, 39jca 304 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → (𝑦 <Q 𝑧𝑧 <Q 𝑦))
4140ex 114 . . . . 5 ((𝐴<P 𝐵𝑞Q) → (((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 <Q 𝑧𝑧 <Q 𝑦)))
423, 41mtoi 654 . . . 4 ((𝐴<P 𝐵𝑞Q) → ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
4342alrimivv 1863 . . 3 ((𝐴<P 𝐵𝑞Q) → ∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
44 ltexprlem.1 . . . . . . . . . . . 12 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
4544ltexprlemell 7539 . . . . . . . . . . 11 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4644ltexprlemelu 7540 . . . . . . . . . . 11 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
4745, 46anbi12i 456 . . . . . . . . . 10 ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
48 anandi 580 . . . . . . . . . 10 ((𝑞Q ∧ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
4947, 48bitr4i 186 . . . . . . . . 9 ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (𝑞Q ∧ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
5049baib 909 . . . . . . . 8 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
51 eleq1 2229 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
52 oveq1 5849 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦 +Q 𝑞) = (𝑧 +Q 𝑞))
5352eleq1d 2235 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦 +Q 𝑞) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))
5451, 53anbi12d 465 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
5554cbvexv 1906 . . . . . . . . 9 (∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))
5655anbi2i 453 . . . . . . . 8 ((∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
5750, 56bitrdi 195 . . . . . . 7 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
58 eeanv 1920 . . . . . . 7 (∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
5957, 58bitr4di 197 . . . . . 6 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6059notbid 657 . . . . 5 (𝑞Q → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ¬ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
61 alnex 1487 . . . . . . 7 (∀𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ ¬ ∃𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
6261albii 1458 . . . . . 6 (∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ ∀𝑦 ¬ ∃𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
63 alnex 1487 . . . . . 6 (∀𝑦 ¬ ∃𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ ¬ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
6462, 63bitri 183 . . . . 5 (∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) ↔ ¬ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
6560, 64bitr4di 197 . . . 4 (𝑞Q → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6665adantl 275 . . 3 ((𝐴<P 𝐵𝑞Q) → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6743, 66mpbird 166 . 2 ((𝐴<P 𝐵𝑞Q) → ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
6867ralrimiva 2539 1 (𝐴<P 𝐵 → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3a 968  wal 1341   = wceq 1343  wex 1480  wcel 2136  wral 2444  {crab 2448  cop 3579   class class class wbr 3982  cfv 5188  (class class class)co 5842  1st c1st 6106  2nd c2nd 6107  Qcnq 7221   +Q cplq 7223   <Q cltq 7226  Pcnp 7232  <P cltp 7236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-ltnqqs 7294  df-inp 7407  df-iltp 7411
This theorem is referenced by:  ltexprlempr  7549
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