New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > cnvsn | GIF version |
Description: Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) |
Ref | Expression |
---|---|
cnvsn.1 | ⊢ A ∈ V |
cnvsn.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
cnvsn | ⊢ ◡{〈A, B〉} = {〈B, A〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . . . 6 ⊢ y ∈ V | |
2 | vex 2862 | . . . . . 6 ⊢ x ∈ V | |
3 | 1, 2 | opex 4588 | . . . . 5 ⊢ 〈y, x〉 ∈ V |
4 | 3 | elsnc 3756 | . . . 4 ⊢ (〈y, x〉 ∈ {〈A, B〉} ↔ 〈y, x〉 = 〈A, B〉) |
5 | ancom 437 | . . . . 5 ⊢ ((y = A ∧ x = B) ↔ (x = B ∧ y = A)) | |
6 | opth 4602 | . . . . 5 ⊢ (〈y, x〉 = 〈A, B〉 ↔ (y = A ∧ x = B)) | |
7 | opth 4602 | . . . . 5 ⊢ (〈x, y〉 = 〈B, A〉 ↔ (x = B ∧ y = A)) | |
8 | 5, 6, 7 | 3bitr4i 268 | . . . 4 ⊢ (〈y, x〉 = 〈A, B〉 ↔ 〈x, y〉 = 〈B, A〉) |
9 | 4, 8 | bitri 240 | . . 3 ⊢ (〈y, x〉 ∈ {〈A, B〉} ↔ 〈x, y〉 = 〈B, A〉) |
10 | opelcnv 4893 | . . 3 ⊢ (〈x, y〉 ∈ ◡{〈A, B〉} ↔ 〈y, x〉 ∈ {〈A, B〉}) | |
11 | 2, 1 | opex 4588 | . . . 4 ⊢ 〈x, y〉 ∈ V |
12 | 11 | elsnc 3756 | . . 3 ⊢ (〈x, y〉 ∈ {〈B, A〉} ↔ 〈x, y〉 = 〈B, A〉) |
13 | 9, 10, 12 | 3bitr4i 268 | . 2 ⊢ (〈x, y〉 ∈ ◡{〈A, B〉} ↔ 〈x, y〉 ∈ {〈B, A〉}) |
14 | 13 | eqrelriv 4850 | 1 ⊢ ◡{〈A, B〉} = {〈B, A〉} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 Vcvv 2859 {csn 3737 〈cop 4561 ◡ccnv 4771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-cnv 4785 |
This theorem is referenced by: opswap 5074 f1osn 5322 |
Copyright terms: Public domain | W3C validator |