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Theorem ltexprlemdisj 7421
Description: Our constructed difference is disjoint. Lemma for ltexpri 7428. (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 7213 . . . . . 6 <Q Or Q
2 ltrelnq 7180 . . . . . 6 <Q ⊆ (Q × Q)
31, 2son2lpi 4935 . . . . 5 ¬ (𝑦 <Q 𝑧𝑧 <Q 𝑦)
4 ltrelpr 7320 . . . . . . . . . . . . . . . 16 <P ⊆ (P × P)
54brel 4591 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → (𝐴P𝐵P))
65simprd 113 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
7 prop 7290 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
86, 7syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 prltlu 7302 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
108, 9syl3an1 1249 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
11103expb 1182 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1211adantlr 468 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 +Q 𝑞) ∈ (1st𝐵) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1312adantrll 475 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑞))
1413adantrrl 477 . . . . . . . 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 7215 . . . . . . . . . 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 7290 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1917, 18syl 14 . . . . . . . . . . . 12 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
20 elprnqu 7297 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2119, 20sylan 281 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2221ad2ant2r 500 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) → 𝑦Q)
2322adantrr 470 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦Q)
24 elprnql 7296 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
2519, 24sylan 281 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
2625ad2ant2r 500 . . . . . . . . . 10 (((𝐴<P 𝐵𝑞Q) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))) → 𝑧Q)
2726adantrl 469 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑧Q)
28 simplr 519 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑞Q)
29 addcomnqg 7196 . . . . . . . . . 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 5940 . . . . . . . 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 7302 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑧 <Q 𝑦)
3419, 33syl3an1 1249 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑧 <Q 𝑦)
35343com23 1187 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧 <Q 𝑦)
36353expb 1182 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3736adantlr 468 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3837adantrlr 476 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ 𝑧 ∈ (1st𝐴))) → 𝑧 <Q 𝑦)
3938adantrrr 478 . . . . . . 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 653 . . . 4 ((𝐴<P 𝐵𝑞Q) → ¬ ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
4342alrimivv 1847 . . 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 7413 . . . . . . . . . . 11 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4644ltexprlemelu 7414 . . . . . . . . . . 11 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
4745, 46anbi12i 455 . . . . . . . . . 10 ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ((𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ∧ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
48 anandi 579 . . . . . . . . . 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 904 . . . . . . . 8 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
51 eleq1 2202 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
52 oveq1 5781 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦 +Q 𝑞) = (𝑧 +Q 𝑞))
5352eleq1d 2208 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦 +Q 𝑞) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))
5451, 53anbi12d 464 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵))))
5554cbvexv 1890 . . . . . . . . 9 (∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))
5655anbi2i 452 . . . . . . . 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 195 . . . . . . 7 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
58 eeanv 1904 . . . . . . 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 197 . . . . . 6 (𝑞Q → ((𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
6059notbid 656 . . . . 5 (𝑞Q → (¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ↔ ¬ ∃𝑦𝑧((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ∧ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑞) ∈ (2nd𝐵)))))
61 alnex 1475 . . . . . . 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 1446 . . . . . 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 1475 . . . . . 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, 64syl6bbr 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 2505 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 962  wal 1329   = wceq 1331  wex 1468  wcel 1480  wral 2416  {crab 2420  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095   +Q cplq 7097   <Q cltq 7100  Pcnp 7106  <P cltp 7110
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-ltnqqs 7168  df-inp 7281  df-iltp 7285
This theorem is referenced by:  ltexprlempr  7423
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