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Theorem ltexprlemdisj 7068
Description: Our constructed difference is disjoint. Lemma for ltexpri 7075. (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 6860 . . . . . 6 <Q Or Q
2 ltrelnq 6827 . . . . . 6 <Q ⊆ (Q × Q)
31, 2son2lpi 4783 . . . . 5 ¬ (𝑦 <Q 𝑧𝑧 <Q 𝑦)
4 ltrelpr 6967 . . . . . . . . . . . . . . . 16 <P ⊆ (P × P)
54brel 4448 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → (𝐴P𝐵P))
65simprd 112 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
7 prop 6937 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
86, 7syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 prltlu 6949 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
108, 9syl3an1 1203 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
11103expb 1140 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1211adantlr 461 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1312adantrll 468 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1413adantrrl 470 . . . . . . . 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 6862 . . . . . . . . . 10 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
1615adantl 271 . . . . . . . . 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 110 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐴P)
18 prop 6937 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1917, 18syl 14 . . . . . . . . . . . 12 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
20 elprnqu 6944 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2119, 20sylan 277 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2221ad2ant2r 493 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) → 𝑦Q)
2322adantrr 463 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦Q)
24 elprnql 6943 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
2519, 24sylan 277 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
2625ad2ant2r 493 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → 𝑧Q)
2726adantrl 462 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑧Q)
28 simplr 497 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑞Q)
29 addcomnqg 6843 . . . . . . . . . 10 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 271 . . . . . . . . 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 5749 . . . . . . . 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 165 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦 <Q 𝑧)
33 prltlu 6949 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑧 <Q 𝑦)
3419, 33syl3an1 1203 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑧 <Q 𝑦)
35343com23 1145 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧 <Q 𝑦)
36353expb 1140 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3736adantlr 461 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3837adantrlr 469 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3938adantrrr 471 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑧 <Q 𝑦)
4032, 39jca 300 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → (𝑦 <Q 𝑧𝑧 <Q 𝑦))
4140ex 113 . . . . 5 ((𝐴<P 𝐵𝑞Q) → (((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 <Q 𝑧𝑧 <Q 𝑦)))
423, 41mtoi 623 . . . 4 ((𝐴<P 𝐵𝑞Q) → ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
4342alrimivv 1798 . . 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 7060 . . . . . . . . . . 11 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4644ltexprlemelu 7061 . . . . . . . . . . 11 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
4745, 46anbi12i 448 . . . . . . . . . 10 ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
48 anandi 555 . . . . . . . . . 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 185 . . . . . . . . 9 ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (𝑞Q ∧ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
5049baib 862 . . . . . . . 8 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
51 eleq1 2145 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
52 oveq1 5598 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦 +Q 𝑞) = (𝑧 +Q 𝑞))
5352eleq1d 2151 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦 +Q 𝑞) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))
5451, 53anbi12d 457 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
5554cbvexv 1838 . . . . . . . . 9 (∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))
5655anbi2i 445 . . . . . . . 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, 56syl6bb 194 . . . . . . 7 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
58 eeanv 1850 . . . . . . 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, 58syl6bbr 196 . . . . . 6 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6059notbid 625 . . . . 5 (𝑞Q → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ¬ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
61 alnex 1429 . . . . . . 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 1400 . . . . . 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 1429 . . . . . 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 182 . . . . 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, 64syl6bbr 196 . . . 4 (𝑞Q → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6665adantl 271 . . 3 ((𝐴<P 𝐵𝑞Q) → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∀𝑦𝑧 ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6743, 66mpbird 165 . 2 ((𝐴<P 𝐵𝑞Q) → ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
6867ralrimiva 2440 1 (𝐴<P 𝐵 → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  w3a 920  wal 1283   = wceq 1285  wex 1422  wcel 1434  wral 2353  {crab 2357  cop 3425   class class class wbr 3811  cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  Qcnq 6742   +Q cplq 6744   <Q cltq 6747  Pcnp 6753  <P cltp 6757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-ltnqqs 6815  df-inp 6928  df-iltp 6932
This theorem is referenced by:  ltexprlempr  7070
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