Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > en2other2 | Unicode version |
Description: Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Ref | Expression |
---|---|
en2other2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2eleq 7019 | . . . . . . 7 | |
2 | prcom 3569 | . . . . . . 7 | |
3 | 1, 2 | syl6eq 2166 | . . . . . 6 |
4 | 3 | difeq1d 3163 | . . . . 5 |
5 | difprsnss 3628 | . . . . 5 | |
6 | 4, 5 | eqsstrdi 3119 | . . . 4 |
7 | simpl 108 | . . . . . 6 | |
8 | 1onn 6384 | . . . . . . . . . 10 | |
9 | 8 | a1i 9 | . . . . . . . . 9 |
10 | simpr 109 | . . . . . . . . . 10 | |
11 | df-2o 6282 | . . . . . . . . . 10 | |
12 | 10, 11 | breqtrdi 3939 | . . . . . . . . 9 |
13 | dif1en 6741 | . . . . . . . . 9 | |
14 | 9, 12, 7, 13 | syl3anc 1201 | . . . . . . . 8 |
15 | en1uniel 6666 | . . . . . . . 8 | |
16 | eldifsni 3622 | . . . . . . . 8 | |
17 | 14, 15, 16 | 3syl 17 | . . . . . . 7 |
18 | 17 | necomd 2371 | . . . . . 6 |
19 | eldifsn 3620 | . . . . . 6 | |
20 | 7, 18, 19 | sylanbrc 413 | . . . . 5 |
21 | 20 | snssd 3635 | . . . 4 |
22 | 6, 21 | eqssd 3084 | . . 3 |
23 | 22 | unieqd 3717 | . 2 |
24 | unisng 3723 | . . 3 | |
25 | 24 | adantr 274 | . 2 |
26 | 23, 25 | eqtrd 2150 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wne 2285 cdif 3038 csn 3497 cpr 3498 cuni 3706 class class class wbr 3899 csuc 4257 com 4474 c1o 6274 c2o 6275 cen 6600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1o 6281 df-2o 6282 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |