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Mirrors > Home > MPE Home > Th. List > unexd | Structured version Visualization version GIF version |
Description: The union of two sets is a set. (Contributed by SN, 16-Jul-2024.) |
Ref | Expression |
---|---|
unexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
unexd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
unexd | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | unexd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | unexg 7463 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | |
4 | 1, 2, 3 | syl2anc 588 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 Vcvv 3407 ∪ cun 3852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 df-v 3409 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-sn 4516 df-pr 4518 df-uni 4792 |
This theorem is referenced by: sexp2 33333 sexp3 33339 ofun 39702 |
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