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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 4622 | . . 3 ⊢ (𝐴 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = {𝐴}) | |
2 | 1 | unieqd 4820 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = ∪ {𝐴}) |
3 | unisng 4827 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝐴} = 𝐴) | |
4 | 2, 3 | eqtrd 2774 | 1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3058 {csn 4526 ∪ cuni 4806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-un 3858 df-in 3860 df-ss 3870 df-sn 4527 df-pr 4529 df-uni 4807 |
This theorem is referenced by: (None) |
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