Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > opelxp | GIF version |
Description: Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opelxp | ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp2 4363 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉) | |
2 | vex 2560 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | vex 2560 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth2 3977 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ (𝐴 = 𝑥 ∧ 𝐵 = 𝑦)) |
5 | eleq1 2100 | . . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐶 ↔ 𝑥 ∈ 𝐶)) | |
6 | eleq1 2100 | . . . . . . 7 ⊢ (𝐵 = 𝑦 → (𝐵 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)) | |
7 | 5, 6 | bi2anan9 538 | . . . . . 6 ⊢ ((𝐴 = 𝑥 ∧ 𝐵 = 𝑦) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
8 | 4, 7 | sylbi 114 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
9 | 8 | biimprcd 149 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))) |
10 | 9 | rexlimivv 2438 | . . 3 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
11 | eqid 2040 | . . . 4 ⊢ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉 | |
12 | opeq1 3549 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
13 | 12 | eqeq2d 2051 | . . . . 5 ⊢ (𝑥 = 𝐴 → (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐴, 𝑦〉)) |
14 | opeq2 3550 | . . . . . 6 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
15 | 14 | eqeq2d 2051 | . . . . 5 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝐵〉 = 〈𝐴, 𝑦〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉)) |
16 | 13, 15 | rspc2ev 2664 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉) |
17 | 11, 16 | mp3an3 1221 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉) |
18 | 10, 17 | impbii 117 | . 2 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
19 | 1, 18 | bitri 173 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 〈cop 3378 × cxp 4343 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 |
This theorem is referenced by: brxp 4375 opelxpi 4376 opelxp1 4377 opelxp2 4378 opthprc 4391 elxp3 4394 opeliunxp 4395 optocl 4416 xpiindim 4473 opelres 4617 resiexg 4653 codir 4713 qfto 4714 xpmlem 4744 rnxpid 4755 ssrnres 4763 dfco2 4820 relssdmrn 4841 ressn 4858 opelf 5062 fnovex 5538 oprab4 5575 resoprab 5597 elmpt2cl 5698 fo1stresm 5788 fo2ndresm 5789 dfoprab4 5818 xporderlem 5852 brecop 6196 xpdom2 6305 enq0enq 6529 enq0sym 6530 enq0tr 6532 nqnq0pi 6536 nnnq0lem1 6544 elinp 6572 genipv 6607 prsrlem1 6827 gt0srpr 6833 opelcn 6903 opelreal 6904 elreal2 6907 frecuzrdgrrn 9194 frec2uzrdg 9195 frecuzrdgrom 9196 frecuzrdgsuc 9201 |
Copyright terms: Public domain | W3C validator |