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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 6692 | . . 3 | |
2 | bren 6692 | . . 3 | |
3 | eeanv 1912 | . . . 4 | |
4 | vex 2715 | . . . . . . . 8 | |
5 | vex 2715 | . . . . . . . 8 | |
6 | 4, 5 | unex 4401 | . . . . . . 7 |
7 | f1oun 5434 | . . . . . . 7 | |
8 | f1oen3g 6699 | . . . . . . 7 | |
9 | 6, 7, 8 | sylancr 411 | . . . . . 6 |
10 | 9 | ex 114 | . . . . 5 |
11 | 10 | exlimivv 1876 | . . . 4 |
12 | 3, 11 | sylbir 134 | . . 3 |
13 | 1, 2, 12 | syl2anb 289 | . 2 |
14 | 13 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wex 1472 wcel 2128 cvv 2712 cun 3100 cin 3101 c0 3394 class class class wbr 3965 wf1o 5169 cen 6683 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-en 6686 |
This theorem is referenced by: enpr2d 6762 phplem2 6798 fiunsnnn 6826 unsnfi 6863 endjusym 7040 pm54.43 7125 endjudisj 7145 djuen 7146 frecfzennn 10325 unennn 12137 |
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