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Mirrors > Home > NFE Home > Th. List > uni1exg | Unicode version |
Description: The unit union operator preserves sethood. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
uni1exg | ⋃1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni12 4292 | . 2 ⋃1 P6 k | |
2 | vvex 4110 | . . . 4 | |
3 | xpkexg 4289 | . . . 4 k | |
4 | 2, 3 | mpan 651 | . . 3 k |
5 | p6exg 4291 | . . 3 k P6 k | |
6 | 4, 5 | syl 15 | . 2 P6 k |
7 | 1, 6 | syl5eqel 2437 | 1 ⋃1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1710 cvv 2860 ⋃1cuni1 4134 k cxpk 4175 P6 cp6 4179 |
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 4079 ax-xp 4080 ax-cnv 4081 ax-typlower 4087 ax-sn 4088 |
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 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-ss 3260 df-nul 3552 df-sn 3742 df-pr 3743 df-uni 3893 df-opk 4059 df-1c 4137 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-p6 4192 |
This theorem is referenced by: uni1ex 4294 uniexg 4317 intexg 4320 coexg 4750 siexg 4753 |
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