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Mirrors > Home > ILE Home > Th. List > ltdfpr | GIF version |
Description: More convenient form of df-iltp 7457. (Contributed by Jim Kingdon, 15-Dec-2019.) |
Ref | Expression |
---|---|
ltdfpr | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4001 | . . 3 ⊢ (𝐴<P 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ <P ) | |
2 | df-iltp 7457 | . . . 4 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} | |
3 | 2 | eleq2i 2244 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ <P ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}) |
4 | 1, 3 | bitri 184 | . 2 ⊢ (𝐴<P 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}) |
5 | simpl 109 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴) | |
6 | 5 | fveq2d 5515 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (2nd ‘𝑥) = (2nd ‘𝐴)) |
7 | 6 | eleq2d 2247 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑞 ∈ (2nd ‘𝑥) ↔ 𝑞 ∈ (2nd ‘𝐴))) |
8 | simpr 110 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
9 | 8 | fveq2d 5515 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (1st ‘𝑦) = (1st ‘𝐵)) |
10 | 9 | eleq2d 2247 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑞 ∈ (1st ‘𝑦) ↔ 𝑞 ∈ (1st ‘𝐵))) |
11 | 7, 10 | anbi12d 473 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)) ↔ (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) |
12 | 11 | rexbidv 2478 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) |
13 | 12 | opelopab2a 4262 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) |
14 | 4, 13 | bitrid 192 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 〈cop 3594 class class class wbr 4000 {copab 4060 ‘cfv 5212 1st c1st 6133 2nd c2nd 6134 Qcnq 7267 Pcnp 7278 <P cltp 7282 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-iota 5174 df-fv 5220 df-iltp 7457 |
This theorem is referenced by: nqprl 7538 nqpru 7539 ltprordil 7576 ltnqpr 7580 ltnqpri 7581 ltpopr 7582 ltsopr 7583 ltaddpr 7584 ltexprlemm 7587 ltexprlemopu 7590 ltexprlemru 7599 aptiprleml 7626 aptiprlemu 7627 archpr 7630 cauappcvgprlem2 7647 caucvgprlem2 7667 caucvgprprlemopu 7686 caucvgprprlemexbt 7693 caucvgprprlem2 7697 suplocexprlemloc 7708 suplocexprlemub 7710 suplocexprlemlub 7711 |
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