Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltexprlemm GIF version

Theorem ltexprlemm 6698
 Description: Our constructed difference is inhabited. Lemma for ltexpri 6711. (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 6603 . . . . . . . . 9 <P ⊆ (P × P)
21brel 4392 . . . . . . . 8 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltdfpr 6604 . . . . . . . . 9 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))))
43biimpd 132 . . . . . . . 8 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))))
52, 4mpcom 32 . . . . . . 7 (𝐴<P 𝐵 → ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))
6 simprrl 491 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → 𝑦 ∈ (2nd𝐴))
72simprd 107 . . . . . . . . . . . . 13 (𝐴<P 𝐵𝐵P)
8 prop 6573 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 prnmaxl 6586 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤)
108, 9sylan 267 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤 ∈ (1st𝐵)𝑦 <Q 𝑤)
11 ltexnqi 6507 . . . . . . . . . . . . . . . . . 18 (𝑦 <Q 𝑤 → ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤)
1211reximi 2416 . . . . . . . . . . . . . . . . 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 2312 . . . . . . . . . . . . . . . 16 (∃𝑤 ∈ (1st𝐵)∃𝑞Q (𝑦 +Q 𝑞) = 𝑤 ↔ ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1513, 14sylib 127 . . . . . . . . . . . . . . 15 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
16 r19.42v 2467 . . . . . . . . . . . . . . . 16 (∃𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ (𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1716exbii 1496 . . . . . . . . . . . . . . 15 (∃𝑤𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) ↔ ∃𝑤(𝑤 ∈ (1st𝐵) ∧ ∃𝑞Q (𝑦 +Q 𝑞) = 𝑤))
1815, 17sylibr 137 . . . . . . . . . . . . . 14 ((𝐵P𝑦 ∈ (1st𝐵)) → ∃𝑤𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤))
19 eleq1 2100 . . . . . . . . . . . . . . . . 17 ((𝑦 +Q 𝑞) = 𝑤 → ((𝑦 +Q 𝑞) ∈ (1st𝐵) ↔ 𝑤 ∈ (1st𝐵)))
2019biimparc 283 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → (𝑦 +Q 𝑞) ∈ (1st𝐵))
2120reximi 2416 . . . . . . . . . . . . . . 15 (∃𝑞Q (𝑤 ∈ (1st𝐵) ∧ (𝑦 +Q 𝑞) = 𝑤) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2221exlimiv 1489 . . . . . . . . . . . . . 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 267 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑦 ∈ (1st𝐵)) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2524adantrl 447 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵))) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
2625adantrl 447 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))
276, 26jca 290 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)))) → (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2827expr 357 . . . . . . . 8 ((𝐴<P 𝐵𝑦Q) → ((𝑦 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵))))
2928reximdva 2421 . . . . . . 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 2467 . . . . . . 7 (∃𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3231rexbii 2331 . . . . . 6 (∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ ∃𝑞Q (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3330, 32sylibr 137 . . . . 5 (𝐴<P 𝐵 → ∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
34 rexcom 2474 . . . . 5 (∃𝑦Q𝑞Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
3533, 34sylib 127 . . . 4 (𝐴<P 𝐵 → ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
362simpld 105 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐴P)
37 prop 6573 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
38 elprnqu 6580 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
3937, 38sylan 267 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
4036, 39sylan 267 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
4140ex 108 . . . . . . . . . 10 (𝐴<P 𝐵 → (𝑦 ∈ (2nd𝐴) → 𝑦Q))
4241pm4.71rd 374 . . . . . . . . 9 (𝐴<P 𝐵 → (𝑦 ∈ (2nd𝐴) ↔ (𝑦Q𝑦 ∈ (2nd𝐴))))
4342anbi1d 438 . . . . . . . 8 (𝐴<P 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ((𝑦Q𝑦 ∈ (2nd𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
44 anass 381 . . . . . . . 8 (((𝑦Q𝑦 ∈ (2nd𝐴)) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4543, 44syl6bb 185 . . . . . . 7 (𝐴<P 𝐵 → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ (𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
4645exbidv 1706 . . . . . 6 (𝐴<P 𝐵 → (∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
4746rexbidv 2327 . . . . 5 (𝐴<P 𝐵 → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))))
48 df-rex 2312 . . . . . 6 (∃𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
4948rexbii 2331 . . . . 5 (∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦(𝑦Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5047, 49syl6bbr 187 . . . 4 (𝐴<P 𝐵 → (∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)) ↔ ∃𝑞Q𝑦Q (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5135, 50mpbird 156 . . 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 6696 . . . . 5 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
5453rexbii 2331 . . . 4 (∃𝑞Q 𝑞 ∈ (1st𝐶) ↔ ∃𝑞Q (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
55 ssid 2964 . . . . 5 QQ
56 rexss 3007 . . . . 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 176 . . 3 (∃𝑞Q 𝑞 ∈ (1st𝐶) ↔ ∃𝑞Q𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵)))
5951, 58sylibr 137 . 2 (𝐴<P 𝐵 → ∃𝑞Q 𝑞 ∈ (1st𝐶))
60 nfv 1421 . . 3 𝑟 𝐴<P 𝐵
61 nfre1 2365 . . 3 𝑟𝑟Q 𝑟 ∈ (2nd𝐶)
62 prmu 6576 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
63 rexex 2368 . . . . 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 6580 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
678, 66sylan 267 . . . . . 6 ((𝐵P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
687, 67sylan 267 . . . . 5 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → 𝑟Q)
69 prml 6575 . . . . . . . . 9 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑦Q 𝑦 ∈ (1st𝐴))
7037, 69syl 14 . . . . . . . 8 (𝐴P → ∃𝑦Q 𝑦 ∈ (1st𝐴))
71 rexex 2368 . . . . . . . 8 (∃𝑦Q 𝑦 ∈ (1st𝐴) → ∃𝑦 𝑦 ∈ (1st𝐴))
7236, 70, 713syl 17 . . . . . . 7 (𝐴<P 𝐵 → ∃𝑦 𝑦 ∈ (1st𝐴))
7372adantr 261 . . . . . 6 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑦 𝑦 ∈ (1st𝐴))
74683adant3 924 . . . . . . . . 9 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟Q)
75 simp3 906 . . . . . . . . . 10 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐴))
76 elprnql 6579 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7737, 76sylan 267 . . . . . . . . . . . . . 14 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7836, 77sylan 267 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
79783adant2 923 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦Q)
80 addcomnqg 6479 . . . . . . . . . . . 12 ((𝑟Q𝑦Q) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
8174, 79, 80syl2anc 391 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟 +Q 𝑦) = (𝑦 +Q 𝑟))
82 ltaddnq 6505 . . . . . . . . . . . . 13 ((𝑟Q𝑦Q) → 𝑟 <Q (𝑟 +Q 𝑦))
8374, 79, 82syl2anc 391 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 <Q (𝑟 +Q 𝑦))
84 prcunqu 6583 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
858, 84sylan 267 . . . . . . . . . . . . . 14 ((𝐵P𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
867, 85sylan 267 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → (𝑟 <Q (𝑟 +Q 𝑦) → (𝑟 +Q 𝑦) ∈ (2nd𝐵)))
87863adant3 924 . . . . . . . . . . . 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 2115 . . . . . . . . . 10 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑦 +Q 𝑟) ∈ (2nd𝐵))
90 19.8a 1482 . . . . . . . . . 10 ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
9175, 89, 90syl2anc 391 . . . . . . . . 9 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
9274, 91jca 290 . . . . . . . 8 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
9352ltexprlemelu 6697 . . . . . . . 8 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
9492, 93sylibr 137 . . . . . . 7 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐶))
95943expa 1104 . . . . . 6 (((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) ∧ 𝑦 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐶))
9673, 95exlimddv 1778 . . . . 5 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → 𝑟 ∈ (2nd𝐶))
97 19.8a 1482 . . . . 5 ((𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
9868, 96, 97syl2anc 391 . . . 4 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
99 df-rex 2312 . . . 4 (∃𝑟Q 𝑟 ∈ (2nd𝐶) ↔ ∃𝑟(𝑟Q𝑟 ∈ (2nd𝐶)))
10098, 99sylibr 137 . . 3 ((𝐴<P 𝐵𝑟 ∈ (2nd𝐵)) → ∃𝑟Q 𝑟 ∈ (2nd𝐶))
10160, 61, 65, 100exlimdd 1752 . 2 (𝐴<P 𝐵 → ∃𝑟Q 𝑟 ∈ (2nd𝐶))
10259, 101jca 290 1 (𝐴<P 𝐵 → (∃𝑞Q 𝑞 ∈ (1st𝐶) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  {crab 2310   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  Qcnq 6378   +Q cplq 6380
 Copyright terms: Public domain W3C validator