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Theorem ltexprlemm 7562
Description: Our constructed difference is inhabited. Lemma for ltexpri 7575. (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 7467 . . . . . . . . 9 <P ⊆ (P × P)
21brel 4663 . . . . . . . 8 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltdfpr 7468 . . . . . . . . 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 534 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → 𝑦 ∈ (2nd𝐴))
72simprd 113 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐵P)
8 prop 7437 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 prnmaxl 7450 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤)
108, 9sylan 281 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤)
11 ltexnqi 7371 . . . . . . . . . . . . . . . . . 18 (𝑦 <Q 𝑤 → ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤)
1211reximi 2567 . . . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . 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 2627 . . . . . . . . . . . . . . . 16 (∃𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ (𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1716exbii 1598 . . . . . . . . . . . . . . 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 2233 . . . . . . . . . . . . . . . . 17 ((𝑦 +Q 𝑞) = 𝑤 → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ↔ 𝑤 ∈ (1st𝐵)))
2019biimparc 297 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → (𝑦 +Q 𝑞) ∈ (1st𝐵))
2120reximi 2567 . . . . . . . . . . . . . . 15 (∃𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2221exlimiv 1591 . . . . . . . . . . . . . 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 281 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑦 ∈ (1st𝐵)) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2524adantrl 475 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2625adantrl 475 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
276, 26jca 304 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2827expr 373 . . . . . . . 8 ((𝐴<P 𝐵𝑦Q) → ((𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))))
2928reximdva 2572 . . . . . . 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 2627 . . . . . . 7 (∃𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3231rexbii 2477 . . . . . 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 2634 . . . . 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 7437 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
38 elprnqu 7444 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
3937, 38sylan 281 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
4036, 39sylan 281 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
4140ex 114 . . . . . . . . . 10 (𝐴<P 𝐵 → (𝑦 ∈ (2nd𝐴) → 𝑦Q))
4241pm4.71rd 392 . . . . . . . . 9 (𝐴<P 𝐵 → (𝑦 ∈ (2nd𝐴) ↔ (𝑦Q𝑦 ∈ (2nd𝐴))))
4342anbi1d 462 . . . . . . . 8 (𝐴<P 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ((𝑦Q𝑦 ∈ (2nd𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
44 anass 399 . . . . . . . 8 (((𝑦Q𝑦 ∈ (2nd𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4543, 44bitrdi 195 . . . . . . 7 (𝐴<P 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
4645exbidv 1818 . . . . . 6 (𝐴<P 𝐵 → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
4746rexbidv 2471 . . . . 5 (𝐴<P 𝐵 → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
48 df-rex 2454 . . . . . 6 (∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4948rexbii 2477 . . . . 5 (∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5047, 49bitr4di 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 7560 . . . . 5 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5453rexbii 2477 . . . 4 (∃𝑞Q 𝑞 ∈ (1st𝐶) ↔ ∃𝑞Q (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
55 ssid 3167 . . . . 5 QQ
56 rexss 3214 . . . . 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 186 . . 3 (∃𝑞Q 𝑞 ∈ (1st𝐶) ↔ ∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
5951, 58sylibr 133 . 2 (𝐴<P 𝐵 → ∃𝑞Q 𝑞 ∈ (1st𝐶))
60 nfv 1521 . . 3 𝑟 𝐴<P 𝐵
61 nfre1 2513 . . 3 𝑟𝑟Q 𝑟 ∈ (2nd𝐶)
62 prmu 7440 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
63 rexex 2516 . . . . 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 7444 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
678, 66sylan 281 . . . . . 6 ((𝐵P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
687, 67sylan 281 . . . . 5 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → 𝑟Q)
69 prml 7439 . . . . . . . . 9 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑦Q 𝑦 ∈ (1st𝐴))
7037, 69syl 14 . . . . . . . 8 (𝐴P → ∃𝑦Q 𝑦 ∈ (1st𝐴))
71 rexex 2516 . . . . . . . 8 (∃𝑦Q 𝑦 ∈ (1st𝐴) → ∃𝑦 𝑦 ∈ (1st𝐴))
7236, 70, 713syl 17 . . . . . . 7 (𝐴<P 𝐵 → ∃𝑦 𝑦 ∈ (1st𝐴))
7372adantr 274 . . . . . 6 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑦 𝑦 ∈ (1st𝐴))
74683adant3 1012 . . . . . . . . 9 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟Q)
75 simp3 994 . . . . . . . . . 10 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐴))
76 elprnql 7443 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7737, 76sylan 281 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7836, 77sylan 281 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
79783adant2 1011 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦Q)
80 addcomnqg 7343 . . . . . . . . . . . 12 ((𝑟Q𝑦Q) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
8174, 79, 80syl2anc 409 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
82 ltaddnq 7369 . . . . . . . . . . . . 13 ((𝑟Q𝑦Q) → 𝑟 <Q (𝑟 +Q 𝑦))
8374, 79, 82syl2anc 409 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 <Q (𝑟 +Q 𝑦))
84 prcunqu 7447 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
858, 84sylan 281 . . . . . . . . . . . . . 14 ((𝐵P𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
867, 85sylan 281 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
87863adant3 1012 . . . . . . . . . . . 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 2248 . . . . . . . . . 10 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑦 +Q 𝑟) ∈ (2nd𝐵))
90 19.8a 1583 . . . . . . . . . 10 ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
9175, 89, 90syl2anc 409 . . . . . . . . 9 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
9274, 91jca 304 . . . . . . . 8 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
9352ltexprlemelu 7561 . . . . . . . 8 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
9492, 93sylibr 133 . . . . . . 7 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐶))
95943expa 1198 . . . . . 6 (((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐶))
9673, 95exlimddv 1891 . . . . 5 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → 𝑟 ∈ (2nd𝐶))
97 19.8a 1583 . . . . 5 ((𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
9868, 96, 97syl2anc 409 . . . 4 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
99 df-rex 2454 . . . 4 (∃𝑟Q 𝑟 ∈ (2nd𝐶) ↔ ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
10098, 99sylibr 133 . . 3 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑟Q 𝑟 ∈ (2nd𝐶))
10160, 61, 65, 100exlimdd 1865 . 2 (𝐴<P 𝐵 → ∃𝑟Q 𝑟 ∈ (2nd𝐶))
10259, 101jca 304 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 973   = wceq 1348  wex 1485  wcel 2141  wrex 2449  {crab 2452  wss 3121  cop 3586   class class class wbr 3989  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  Qcnq 7242   +Q cplq 7244   <Q cltq 7247  Pcnp 7253  <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-ltnqqs 7315  df-inp 7428  df-iltp 7432
This theorem is referenced by:  ltexprlempr  7570
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