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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 7763 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 |
| This theorem is referenced by: sexp2 8171 sexp3 8178 ssltun1 27853 ssltun2 27854 addsproplem2 28003 addsuniflem 28034 ssltmul1 28173 ssltmul2 28174 precsexlem11 28241 suppun2 32693 elrgspnsubrunlem1 33251 elrgspnsubrunlem2 33252 elrgspnsubrun 33253 elrspunsn 33457 ofun 42277 tfsconcatun 43350 rclexi 43628 rtrclexlem 43629 trclubgNEW 43631 cnvrcl0 43638 dfrtrcl5 43642 iunrelexp0 43715 relexpmulg 43723 relexp01min 43726 clnbgrval 47809 |
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