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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 2692 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | elex 2692 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
3 | unexb 4358 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) | |
4 | 3 | biimpi 119 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
5 | 1, 2, 4 | syl2an 287 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 Vcvv 2681 ∪ cun 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-uni 3732 |
This theorem is referenced by: tpexg 4360 eldifpw 4393 xpexg 4648 tposexg 6148 tfrlemisucaccv 6215 tfrlemibxssdm 6217 tfrlemibfn 6218 tfr1onlemsucaccv 6231 tfr1onlembxssdm 6233 tfr1onlembfn 6234 tfrcllemsucaccv 6244 tfrcllembxssdm 6246 tfrcllembfn 6247 rdgtfr 6264 rdgruledefgg 6265 rdgivallem 6271 djuex 6921 zfz1isolem1 10576 ennnfonelemp1 11908 setsvalg 11978 setsex 11980 setsslid 11998 strleund 12036 |
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