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Mirrors > Home > NFE Home > Th. List > uniexg | GIF version |
Description: The sum class of a set is a set. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
uniexg | ⊢ (A ∈ V → ∪A ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni3 4315 | . 2 ⊢ ∪A = ⋃1(◡k Sk “k A) | |
2 | ssetkex 4294 | . . . . 5 ⊢ Sk ∈ V | |
3 | 2 | cnvkex 4287 | . . . 4 ⊢ ◡k Sk ∈ V |
4 | imakexg 4299 | . . . 4 ⊢ ((◡k Sk ∈ V ∧ A ∈ V) → (◡k Sk “k A) ∈ V) | |
5 | 3, 4 | mpan 651 | . . 3 ⊢ (A ∈ V → (◡k Sk “k A) ∈ V) |
6 | uni1exg 4292 | . . 3 ⊢ ((◡k Sk “k A) ∈ V → ⋃1(◡k Sk “k A) ∈ V) | |
7 | 5, 6 | syl 15 | . 2 ⊢ (A ∈ V → ⋃1(◡k Sk “k A) ∈ V) |
8 | 1, 7 | syl5eqel 2437 | 1 ⊢ (A ∈ V → ∪A ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 Vcvv 2859 ∪cuni 3891 ⋃1cuni1 4133 ◡kccnvk 4175 “k cimak 4179 Sk cssetk 4183 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-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: uniex 4317 pw1exb 4326 |
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