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Mirrors > Home > ILE Home > Th. List > unen | Unicode version |
Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
unen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 6251 | . . 3 | |
2 | bren 6251 | . . 3 | |
3 | eeanv 1848 | . . . 4 | |
4 | vex 2604 | . . . . . . . 8 | |
5 | vex 2604 | . . . . . . . 8 | |
6 | 4, 5 | unex 4194 | . . . . . . 7 |
7 | f1oun 5166 | . . . . . . 7 | |
8 | f1oen3g 6257 | . . . . . . 7 | |
9 | 6, 7, 8 | sylancr 405 | . . . . . 6 |
10 | 9 | ex 113 | . . . . 5 |
11 | 10 | exlimivv 1817 | . . . 4 |
12 | 3, 11 | sylbir 133 | . . 3 |
13 | 1, 2, 12 | syl2anb 285 | . 2 |
14 | 13 | imp 122 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 cvv 2601 cun 2971 cin 2972 c0 3251 class class class wbr 3785 wf1o 4921 cen 6242 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-en 6245 |
This theorem is referenced by: phplem2 6339 fiunsnnn 6365 unsnfi 6384 pm54.43 6459 frecfzennn 9419 |
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