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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 6228 | . . 3 | |
2 | bren 6228 | . . 3 | |
3 | eeanv 1807 | . . . 4 | |
4 | vex 2560 | . . . . . . . 8 | |
5 | vex 2560 | . . . . . . . 8 | |
6 | 4, 5 | unex 4176 | . . . . . . 7 |
7 | f1oun 5146 | . . . . . . 7 | |
8 | f1oen3g 6234 | . . . . . . 7 | |
9 | 6, 7, 8 | sylancr 393 | . . . . . 6 |
10 | 9 | ex 108 | . . . . 5 |
11 | 10 | exlimivv 1776 | . . . 4 |
12 | 3, 11 | sylbir 125 | . . 3 |
13 | 1, 2, 12 | syl2anb 275 | . 2 |
14 | 13 | imp 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cvv 2557 cun 2915 cin 2916 c0 3224 class class class wbr 3764 wf1o 4901 cen 6219 |
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-in1 544 ax-in2 545 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-13 1404 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 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-en 6222 |
This theorem is referenced by: phplem2 6316 fiunsnnn 6338 frecfzennn 9203 |
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