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Theorem ltexprlemm 7616
Description: Our constructed difference is inhabited. Lemma for ltexpri 7629. (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 7521 . . . . . . . . 9 <P ⊆ (P × P)
21brel 4692 . . . . . . . 8 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltdfpr 7522 . . . . . . . . 9 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))))
43biimpd 144 . . . . . . . 8 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))))
52, 4mpcom 36 . . . . . . 7 (𝐴<P 𝐵 → ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))
6 simprrl 539 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → 𝑦 ∈ (2nd𝐴))
72simprd 114 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐵P)
8 prop 7491 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 prnmaxl 7504 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤)
108, 9sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤)
11 ltexnqi 7425 . . . . . . . . . . . . . . . . . 18 (𝑦 <Q 𝑤 → ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤)
1211reximi 2586 . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . . 16 (∃𝑤 ∈ (1st𝐵)∃𝑞Q (𝑦 +Q 𝑞) = 𝑤 ↔ ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1513, 14sylib 122 . . . . . . . . . . . . . . 15 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
16 r19.42v 2646 . . . . . . . . . . . . . . . 16 (∃𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ (𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1716exbii 1615 . . . . . . . . . . . . . . 15 (∃𝑤𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1815, 17sylibr 134 . . . . . . . . . . . . . 14 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤))
19 eleq1 2251 . . . . . . . . . . . . . . . . 17 ((𝑦 +Q 𝑞) = 𝑤 → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ↔ 𝑤 ∈ (1st𝐵)))
2019biimparc 299 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → (𝑦 +Q 𝑞) ∈ (1st𝐵))
2120reximi 2586 . . . . . . . . . . . . . . 15 (∃𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2221exlimiv 1608 . . . . . . . . . . . . . 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 283 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑦 ∈ (1st𝐵)) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2524adantrl 478 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2625adantrl 478 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
276, 26jca 306 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2827expr 375 . . . . . . . 8 ((𝐴<P 𝐵𝑦Q) → ((𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))))
2928reximdva 2591 . . . . . . 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 2646 . . . . . . 7 (∃𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3231rexbii 2496 . . . . . 6 (∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3330, 32sylibr 134 . . . . 5 (𝐴<P 𝐵 → ∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
34 rexcom 2653 . . . . 5 (∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3533, 34sylib 122 . . . 4 (𝐴<P 𝐵 → ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
362simpld 112 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐴P)
37 prop 7491 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
38 elprnqu 7498 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
3937, 38sylan 283 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
4036, 39sylan 283 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
4140ex 115 . . . . . . . . . 10 (𝐴<P 𝐵 → (𝑦 ∈ (2nd𝐴) → 𝑦Q))
4241pm4.71rd 394 . . . . . . . . 9 (𝐴<P 𝐵 → (𝑦 ∈ (2nd𝐴) ↔ (𝑦Q𝑦 ∈ (2nd𝐴))))
4342anbi1d 465 . . . . . . . 8 (𝐴<P 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ((𝑦Q𝑦 ∈ (2nd𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
44 anass 401 . . . . . . . 8 (((𝑦Q𝑦 ∈ (2nd𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4543, 44bitrdi 196 . . . . . . 7 (𝐴<P 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
4645exbidv 1835 . . . . . 6 (𝐴<P 𝐵 → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
4746rexbidv 2490 . . . . 5 (𝐴<P 𝐵 → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
48 df-rex 2473 . . . . . 6 (∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4948rexbii 2496 . . . . 5 (∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5047, 49bitr4di 198 . . . 4 (𝐴<P 𝐵 → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5135, 50mpbird 167 . . 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 7614 . . . . 5 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5453rexbii 2496 . . . 4 (∃𝑞Q 𝑞 ∈ (1st𝐶) ↔ ∃𝑞Q (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
55 ssid 3189 . . . . 5 QQ
56 rexss 3236 . . . . 5 (QQ → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
5755, 56ax-mp 5 . . . 4 (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5854, 57bitr4i 187 . . 3 (∃𝑞Q 𝑞 ∈ (1st𝐶) ↔ ∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
5951, 58sylibr 134 . 2 (𝐴<P 𝐵 → ∃𝑞Q 𝑞 ∈ (1st𝐶))
60 nfv 1538 . . 3 𝑟 𝐴<P 𝐵
61 nfre1 2532 . . 3 𝑟𝑟Q 𝑟 ∈ (2nd𝐶)
62 prmu 7494 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
63 rexex 2535 . . . . 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 7498 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
678, 66sylan 283 . . . . . 6 ((𝐵P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
687, 67sylan 283 . . . . 5 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → 𝑟Q)
69 prml 7493 . . . . . . . . 9 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑦Q 𝑦 ∈ (1st𝐴))
7037, 69syl 14 . . . . . . . 8 (𝐴P → ∃𝑦Q 𝑦 ∈ (1st𝐴))
71 rexex 2535 . . . . . . . 8 (∃𝑦Q 𝑦 ∈ (1st𝐴) → ∃𝑦 𝑦 ∈ (1st𝐴))
7236, 70, 713syl 17 . . . . . . 7 (𝐴<P 𝐵 → ∃𝑦 𝑦 ∈ (1st𝐴))
7372adantr 276 . . . . . 6 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑦 𝑦 ∈ (1st𝐴))
74683adant3 1018 . . . . . . . . 9 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟Q)
75 simp3 1000 . . . . . . . . . 10 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐴))
76 elprnql 7497 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7737, 76sylan 283 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7836, 77sylan 283 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
79783adant2 1017 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦Q)
80 addcomnqg 7397 . . . . . . . . . . . 12 ((𝑟Q𝑦Q) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
8174, 79, 80syl2anc 411 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
82 ltaddnq 7423 . . . . . . . . . . . . 13 ((𝑟Q𝑦Q) → 𝑟 <Q (𝑟 +Q 𝑦))
8374, 79, 82syl2anc 411 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 <Q (𝑟 +Q 𝑦))
84 prcunqu 7501 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
858, 84sylan 283 . . . . . . . . . . . . . 14 ((𝐵P𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
867, 85sylan 283 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
87863adant3 1018 . . . . . . . . . . . 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 2266 . . . . . . . . . 10 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑦 +Q 𝑟) ∈ (2nd𝐵))
90 19.8a 1600 . . . . . . . . . 10 ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
9175, 89, 90syl2anc 411 . . . . . . . . 9 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
9274, 91jca 306 . . . . . . . 8 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
9352ltexprlemelu 7615 . . . . . . . 8 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
9492, 93sylibr 134 . . . . . . 7 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐶))
95943expa 1204 . . . . . 6 (((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐶))
9673, 95exlimddv 1909 . . . . 5 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → 𝑟 ∈ (2nd𝐶))
97 19.8a 1600 . . . . 5 ((𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
9868, 96, 97syl2anc 411 . . . 4 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
99 df-rex 2473 . . . 4 (∃𝑟Q 𝑟 ∈ (2nd𝐶) ↔ ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
10098, 99sylibr 134 . . 3 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑟Q 𝑟 ∈ (2nd𝐶))
10160, 61, 65, 100exlimdd 1882 . 2 (𝐴<P 𝐵 → ∃𝑟Q 𝑟 ∈ (2nd𝐶))
10259, 101jca 306 1 (𝐴<P 𝐵 → (∃𝑞Q 𝑞 ∈ (1st𝐶) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 979   = wceq 1363  wex 1502  wcel 2159  wrex 2468  {crab 2471  wss 3143  cop 3609   class class class wbr 4017  cfv 5230  (class class class)co 5890  1st c1st 6156  2nd c2nd 6157  Qcnq 7296   +Q cplq 7298   <Q cltq 7301  Pcnp 7307  <P cltp 7311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-eprel 4303  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-irdg 6388  df-1o 6434  df-oadd 6438  df-omul 6439  df-er 6552  df-ec 6554  df-qs 6558  df-ni 7320  df-pli 7321  df-mi 7322  df-lti 7323  df-plpq 7360  df-mpq 7361  df-enq 7363  df-nqqs 7364  df-plqqs 7365  df-mqqs 7366  df-1nqqs 7367  df-ltnqqs 7369  df-inp 7482  df-iltp 7486
This theorem is referenced by:  ltexprlempr  7624
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