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Mirrors > Home > NFE Home > Th. List > unexg | GIF version |
Description: The union of two sets is a set. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
unexg | ⊢ ((A ∈ V ∧ B ∈ W) → (A ∪ B) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-un 3214 | . 2 ⊢ (A ∪ B) = ( ∼ A ⩃ ∼ B) | |
2 | complexg 4099 | . . 3 ⊢ (A ∈ V → ∼ A ∈ V) | |
3 | complexg 4099 | . . 3 ⊢ (B ∈ W → ∼ B ∈ V) | |
4 | ninexg 4097 | . . 3 ⊢ (( ∼ A ∈ V ∧ ∼ B ∈ V) → ( ∼ A ⩃ ∼ B) ∈ V) | |
5 | 2, 3, 4 | syl2an 463 | . 2 ⊢ ((A ∈ V ∧ B ∈ W) → ( ∼ A ⩃ ∼ B) ∈ V) |
6 | 1, 5 | syl5eqel 2437 | 1 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∪ B) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 Vcvv 2859 ⩃ cnin 3204 ∼ ccompl 3205 ∪ cun 3207 |
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 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 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-v 2861 df-nin 3211 df-compl 3212 df-un 3214 |
This theorem is referenced by: symdifexg 4103 unex 4106 imakexg 4299 ncfindi 4475 opexg 4587 cupvalg 5812 fullfunexg 5859 addccan2nclem2 6264 |
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