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Mirrors > Home > NFE Home > Th. List > intexg | Unicode version |
Description: The intersection of a set is a set. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
intexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfint3 4318 | . 2 ∼ ⋃1k ∼ Sk k | |
2 | ssetkex 4294 | . . . . . 6 Sk | |
3 | 2 | complex 4104 | . . . . 5 ∼ Sk |
4 | 3 | cnvkex 4287 | . . . 4 k ∼ Sk |
5 | imakexg 4299 | . . . 4 k ∼ Sk k ∼ Sk k | |
6 | 4, 5 | mpan 651 | . . 3 k ∼ Sk k |
7 | uni1exg 4292 | . . 3 k ∼ Sk k ⋃1k ∼ Sk k | |
8 | complexg 4099 | . . 3 ⋃1k ∼ Sk k ∼ ⋃1k ∼ Sk k | |
9 | 6, 7, 8 | 3syl 18 | . 2 ∼ ⋃1k ∼ Sk k |
10 | 1, 9 | syl5eqel 2437 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1710 cvv 2859 ∼ ccompl 3205 cint 3926 ⋃1cuni1 4133 kccnvk 4175 kcimak 4179 Sk cssetk 4183 |
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 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-typlower 4086 ax-sn 4087 |
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 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-imak 4189 df-p6 4191 df-sik 4192 df-ssetk 4193 |
This theorem is referenced by: intex 4320 |
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