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Theorem ltexprlemm 7256
Description: Our constructed difference is inhabited. Lemma for ltexpri 7269. (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
ltexprlemm (𝐴<P 𝐵 → (∃𝑞Q 𝑞 ∈ (1st𝐶) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemm
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7161 . . . . . . . . 9 <P ⊆ (P × P)
21brel 4519 . . . . . . . 8 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltdfpr 7162 . . . . . . . . 9 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))))
43biimpd 143 . . . . . . . 8 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))))
52, 4mpcom 36 . . . . . . 7 (𝐴<P 𝐵 → ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))
6 simprrl 507 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → 𝑦 ∈ (2nd𝐴))
72simprd 113 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐵P)
8 prop 7131 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 prnmaxl 7144 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤)
108, 9sylan 278 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤)
11 ltexnqi 7065 . . . . . . . . . . . . . . . . . 18 (𝑦 <Q 𝑤 → ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤)
1211reximi 2482 . . . . . . . . . . . . . . . . 17 (∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤 → ∃𝑤 ∈ (1st𝐵)∃𝑞Q (𝑦 +Q 𝑞) = 𝑤)
1310, 12syl 14 . . . . . . . . . . . . . . . 16 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)∃𝑞Q (𝑦 +Q 𝑞) = 𝑤)
14 df-rex 2376 . . . . . . . . . . . . . . . 16 (∃𝑤 ∈ (1st𝐵)∃𝑞Q (𝑦 +Q 𝑞) = 𝑤 ↔ ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1513, 14sylib 121 . . . . . . . . . . . . . . 15 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
16 r19.42v 2538 . . . . . . . . . . . . . . . 16 (∃𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ (𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1716exbii 1548 . . . . . . . . . . . . . . 15 (∃𝑤𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1815, 17sylibr 133 . . . . . . . . . . . . . 14 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤))
19 eleq1 2157 . . . . . . . . . . . . . . . . 17 ((𝑦 +Q 𝑞) = 𝑤 → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ↔ 𝑤 ∈ (1st𝐵)))
2019biimparc 294 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → (𝑦 +Q 𝑞) ∈ (1st𝐵))
2120reximi 2482 . . . . . . . . . . . . . . 15 (∃𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2221exlimiv 1541 . . . . . . . . . . . . . 14 (∃𝑤𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2318, 22syl 14 . . . . . . . . . . . . 13 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
247, 23sylan 278 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑦 ∈ (1st𝐵)) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2524adantrl 463 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2625adantrl 463 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
276, 26jca 301 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2827expr 368 . . . . . . . 8 ((𝐴<P 𝐵𝑦Q) → ((𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))))
2928reximdva 2487 . . . . . . 7 (𝐴<P 𝐵 → (∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)) → ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))))
305, 29mpd 13 . . . . . 6 (𝐴<P 𝐵 → ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
31 r19.42v 2538 . . . . . . 7 (∃𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3231rexbii 2396 . . . . . 6 (∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3330, 32sylibr 133 . . . . 5 (𝐴<P 𝐵 → ∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
34 rexcom 2545 . . . . 5 (∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3533, 34sylib 121 . . . 4 (𝐴<P 𝐵 → ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
362simpld 111 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐴P)
37 prop 7131 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
38 elprnqu 7138 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
3937, 38sylan 278 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
4036, 39sylan 278 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
4140ex 114 . . . . . . . . . 10 (𝐴<P 𝐵 → (𝑦 ∈ (2nd𝐴) → 𝑦Q))
4241pm4.71rd 387 . . . . . . . . 9 (𝐴<P 𝐵 → (𝑦 ∈ (2nd𝐴) ↔ (𝑦Q𝑦 ∈ (2nd𝐴))))
4342anbi1d 454 . . . . . . . 8 (𝐴<P 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ((𝑦Q𝑦 ∈ (2nd𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
44 anass 394 . . . . . . . 8 (((𝑦Q𝑦 ∈ (2nd𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4543, 44syl6bb 195 . . . . . . 7 (𝐴<P 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
4645exbidv 1760 . . . . . 6 (𝐴<P 𝐵 → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
4746rexbidv 2392 . . . . 5 (𝐴<P 𝐵 → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
48 df-rex 2376 . . . . . 6 (∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4948rexbii 2396 . . . . 5 (∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5047, 49syl6bbr 197 . . . 4 (𝐴<P 𝐵 → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5135, 50mpbird 166 . . 3 (𝐴<P 𝐵 → ∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
52 ltexprlem.1 . . . . . 6 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
5352ltexprlemell 7254 . . . . 5 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5453rexbii 2396 . . . 4 (∃𝑞Q 𝑞 ∈ (1st𝐶) ↔ ∃𝑞Q (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
55 ssid 3059 . . . . 5 QQ
56 rexss 3103 . . . . 5 (QQ → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
5755, 56ax-mp 7 . . . 4 (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5854, 57bitr4i 186 . . 3 (∃𝑞Q 𝑞 ∈ (1st𝐶) ↔ ∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
5951, 58sylibr 133 . 2 (𝐴<P 𝐵 → ∃𝑞Q 𝑞 ∈ (1st𝐶))
60 nfv 1473 . . 3 𝑟 𝐴<P 𝐵
61 nfre1 2430 . . 3 𝑟𝑟Q 𝑟 ∈ (2nd𝐶)
62 prmu 7134 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
63 rexex 2433 . . . . 5 (∃𝑟Q 𝑟 ∈ (2nd𝐵) → ∃𝑟 𝑟 ∈ (2nd𝐵))
6462, 63syl 14 . . . 4 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑟 𝑟 ∈ (2nd𝐵))
657, 8, 643syl 17 . . 3 (𝐴<P 𝐵 → ∃𝑟 𝑟 ∈ (2nd𝐵))
66 elprnqu 7138 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
678, 66sylan 278 . . . . . 6 ((𝐵P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
687, 67sylan 278 . . . . 5 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → 𝑟Q)
69 prml 7133 . . . . . . . . 9 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑦Q 𝑦 ∈ (1st𝐴))
7037, 69syl 14 . . . . . . . 8 (𝐴P → ∃𝑦Q 𝑦 ∈ (1st𝐴))
71 rexex 2433 . . . . . . . 8 (∃𝑦Q 𝑦 ∈ (1st𝐴) → ∃𝑦 𝑦 ∈ (1st𝐴))
7236, 70, 713syl 17 . . . . . . 7 (𝐴<P 𝐵 → ∃𝑦 𝑦 ∈ (1st𝐴))
7372adantr 271 . . . . . 6 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑦 𝑦 ∈ (1st𝐴))
74683adant3 966 . . . . . . . . 9 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟Q)
75 simp3 948 . . . . . . . . . 10 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐴))
76 elprnql 7137 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7737, 76sylan 278 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7836, 77sylan 278 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
79783adant2 965 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦Q)
80 addcomnqg 7037 . . . . . . . . . . . 12 ((𝑟Q𝑦Q) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
8174, 79, 80syl2anc 404 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
82 ltaddnq 7063 . . . . . . . . . . . . 13 ((𝑟Q𝑦Q) → 𝑟 <Q (𝑟 +Q 𝑦))
8374, 79, 82syl2anc 404 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 <Q (𝑟 +Q 𝑦))
84 prcunqu 7141 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
858, 84sylan 278 . . . . . . . . . . . . . 14 ((𝐵P𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
867, 85sylan 278 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
87863adant3 966 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
8883, 87mpd 13 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟 +Q 𝑦) ∈ (2nd𝐵))
8981, 88eqeltrrd 2172 . . . . . . . . . 10 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑦 +Q 𝑟) ∈ (2nd𝐵))
90 19.8a 1534 . . . . . . . . . 10 ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
9175, 89, 90syl2anc 404 . . . . . . . . 9 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
9274, 91jca 301 . . . . . . . 8 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
9352ltexprlemelu 7255 . . . . . . . 8 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
9492, 93sylibr 133 . . . . . . 7 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐶))
95943expa 1146 . . . . . 6 (((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐶))
9673, 95exlimddv 1833 . . . . 5 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → 𝑟 ∈ (2nd𝐶))
97 19.8a 1534 . . . . 5 ((𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
9868, 96, 97syl2anc 404 . . . 4 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
99 df-rex 2376 . . . 4 (∃𝑟Q 𝑟 ∈ (2nd𝐶) ↔ ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
10098, 99sylibr 133 . . 3 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑟Q 𝑟 ∈ (2nd𝐶))
10160, 61, 65, 100exlimdd 1807 . 2 (𝐴<P 𝐵 → ∃𝑟Q 𝑟 ∈ (2nd𝐶))
10259, 101jca 301 1 (𝐴<P 𝐵 → (∃𝑞Q 𝑞 ∈ (1st𝐶) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 927   = wceq 1296  wex 1433  wcel 1445  wrex 2371  {crab 2374  wss 3013  cop 3469   class class class wbr 3867  cfv 5049  (class class class)co 5690  1st c1st 5947  2nd c2nd 5948  Qcnq 6936   +Q cplq 6938   <Q cltq 6941  Pcnp 6947  <P cltp 6951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-ltnqqs 7009  df-inp 7122  df-iltp 7126
This theorem is referenced by:  ltexprlempr  7264
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