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Mirrors > Home > ILE Home > Th. List > unexg | GIF version |
Description: A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) |
Ref | Expression |
---|---|
unexg | ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (A ∪ B) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2560 | . 2 ⊢ (A ∈ 𝑉 → A ∈ V) | |
2 | elex 2560 | . 2 ⊢ (B ∈ 𝑊 → B ∈ V) | |
3 | unexb 4143 | . . 3 ⊢ ((A ∈ V ∧ B ∈ V) ↔ (A ∪ B) ∈ V) | |
4 | 3 | biimpi 113 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ∪ B) ∈ V) |
5 | 1, 2, 4 | syl2an 273 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (A ∪ B) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 Vcvv 2551 ∪ cun 2909 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-uni 3572 |
This theorem is referenced by: tpexg 4145 eldifpw 4174 xpexg 4395 tposexg 5814 tfrlemisucaccv 5880 tfrlemibxssdm 5882 tfrlemibfn 5883 rdgtfr 5901 rdgruledefgg 5902 rdgivallem 5908 |
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