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Mirrors > Home > ILE Home > Th. List > ltdfpr | GIF version |
Description: More convenient form of df-iltp 6453. (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 3756 | . . 3 ⊢ (A<P B ↔ 〈A, B〉 ∈ <P ) | |
2 | df-iltp 6453 | . . . 4 ⊢ <P = {〈x, y〉 ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))} | |
3 | 2 | eleq2i 2101 | . . 3 ⊢ (〈A, B〉 ∈ <P ↔ 〈A, B〉 ∈ {〈x, y〉 ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))}) |
4 | 1, 3 | bitri 173 | . 2 ⊢ (A<P B ↔ 〈A, B〉 ∈ {〈x, y〉 ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))}) |
5 | simpl 102 | . . . . . . 7 ⊢ ((x = A ∧ y = B) → x = A) | |
6 | 5 | fveq2d 5125 | . . . . . 6 ⊢ ((x = A ∧ y = B) → (2nd ‘x) = (2nd ‘A)) |
7 | 6 | eleq2d 2104 | . . . . 5 ⊢ ((x = A ∧ y = B) → (𝑞 ∈ (2nd ‘x) ↔ 𝑞 ∈ (2nd ‘A))) |
8 | simpr 103 | . . . . . . 7 ⊢ ((x = A ∧ y = B) → y = B) | |
9 | 8 | fveq2d 5125 | . . . . . 6 ⊢ ((x = A ∧ y = B) → (1st ‘y) = (1st ‘B)) |
10 | 9 | eleq2d 2104 | . . . . 5 ⊢ ((x = A ∧ y = B) → (𝑞 ∈ (1st ‘y) ↔ 𝑞 ∈ (1st ‘B))) |
11 | 7, 10 | anbi12d 442 | . . . 4 ⊢ ((x = A ∧ y = B) → ((𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)) ↔ (𝑞 ∈ (2nd ‘A) ∧ 𝑞 ∈ (1st ‘B)))) |
12 | 11 | rexbidv 2321 | . . 3 ⊢ ((x = A ∧ y = B) → (∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘A) ∧ 𝑞 ∈ (1st ‘B)))) |
13 | 12 | opelopab2a 3993 | . 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 | syl5bb 181 | 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 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∃wrex 2301 〈cop 3370 class class class wbr 3755 {copab 3808 ‘cfv 4845 1st c1st 5707 2nd c2nd 5708 Qcnq 6264 Pcnp 6275 <P cltp 6279 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-iota 4810 df-fv 4853 df-iltp 6453 |
This theorem is referenced by: nqprl 6533 ltprordil 6565 ltpopr 6569 ltsopr 6570 ltaddpr 6571 ltexprlemm 6574 ltexprlemopu 6577 ltexprlemru 6586 aptiprleml 6611 aptiprlemu 6612 archpr 6615 cauappcvgprlem2 6632 caucvgprlem2 6651 |
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