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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 4656 | . . 3 ⊢ (𝐴 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = {𝐴}) | |
2 | 1 | unieqd 4851 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = ∪ {𝐴}) |
3 | unisng 4856 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝐴} = 𝐴) | |
4 | 2, 3 | eqtrd 2856 | 1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 {csn 4566 ∪ cuni 4837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-un 3940 df-in 3942 df-ss 3951 df-sn 4567 df-pr 4569 df-uni 4838 |
This theorem is referenced by: (None) |
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