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Mirrors > Home > MPE Home > Th. List > unisn3 | Structured version Visualization version GIF version |
Description: Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.) |
Ref | Expression |
---|---|
unisn3 | ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabsn 4720 | . . 3 ⊢ (𝐴 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = {𝐴}) | |
2 | 1 | unieqd 4915 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = ∪ {𝐴}) |
3 | unisng 4922 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝐴} = 𝐴) | |
4 | 2, 3 | eqtrd 2766 | 1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3426 {csn 4623 ∪ cuni 4902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 df-sn 4624 df-pr 4626 df-uni 4903 |
This theorem is referenced by: (None) |
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