New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > elswap | Unicode version |
Description: Membership in the Swap function. (Contributed by SF, 6-Jan-2015.) |
Ref | Expression |
---|---|
elswap | Swap |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-swap 4724 | . . . 4 Swap | |
2 | 1 | eleq2i 2417 | . . 3 Swap |
3 | elopab 4696 | . . 3 | |
4 | 2, 3 | bitri 240 | . 2 Swap |
5 | exrot4 1745 | . . 3 | |
6 | 19.42vv 1907 | . . . 4 | |
7 | 6 | 2exbii 1583 | . . 3 |
8 | df-3an 936 | . . . . . . 7 | |
9 | ancom 437 | . . . . . . 7 | |
10 | 8, 9 | bitr2i 241 | . . . . . 6 |
11 | 10 | 2exbii 1583 | . . . . 5 |
12 | vex 2862 | . . . . . . 7 | |
13 | vex 2862 | . . . . . . 7 | |
14 | 12, 13 | opex 4588 | . . . . . 6 |
15 | 13, 12 | opex 4588 | . . . . . 6 |
16 | opeq1 4578 | . . . . . . 7 | |
17 | 16 | eqeq2d 2364 | . . . . . 6 |
18 | opeq2 4579 | . . . . . . 7 | |
19 | 18 | eqeq2d 2364 | . . . . . 6 |
20 | 14, 15, 17, 19 | ceqsex2v 2896 | . . . . 5 |
21 | 11, 20 | bitri 240 | . . . 4 |
22 | 21 | 2exbii 1583 | . . 3 |
23 | 5, 7, 22 | 3bitr3i 266 | . 2 |
24 | 4, 23 | bitri 240 | 1 Swap |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 w3a 934 wex 1541 wceq 1642 wcel 1710 cop 4561 copab 4622 Swap cswap 4718 |
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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-opab 4623 df-swap 4724 |
This theorem is referenced by: dfswap2 4741 |
Copyright terms: Public domain | W3C validator |