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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 4726 | . . 3 ⊢ (𝐴 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = {𝐴}) | |
2 | 1 | unieqd 4921 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = ∪ {𝐴}) |
3 | unisng 4928 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝐴} = 𝐴) | |
4 | 2, 3 | eqtrd 2768 | 1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3429 {csn 4629 ∪ cuni 4908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-un 3952 df-in 3954 df-ss 3964 df-sn 4630 df-pr 4632 df-uni 4909 |
This theorem is referenced by: (None) |
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