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Mirrors > Home > MPE Home > Th. List > unisn2 | Structured version Visualization version GIF version |
Description: A version of unisn 4891 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.) |
Ref | Expression |
---|---|
unisn2 | ⊢ ∪ {𝐴} ∈ {∅, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisng 4890 | . . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
2 | prid2g 4726 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴}) | |
3 | 1, 2 | eqeltrd 2834 | . 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴}) |
4 | snprc 4682 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
5 | 4 | biimpi 215 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
6 | 5 | unieqd 4883 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
7 | uni0 4900 | . . . 4 ⊢ ∪ ∅ = ∅ | |
8 | 0ex 5268 | . . . . 5 ⊢ ∅ ∈ V | |
9 | 8 | prid1 4727 | . . . 4 ⊢ ∅ ∈ {∅, 𝐴} |
10 | 7, 9 | eqeltri 2830 | . . 3 ⊢ ∪ ∅ ∈ {∅, 𝐴} |
11 | 6, 10 | eqeltrdi 2842 | . 2 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴}) |
12 | 3, 11 | pm2.61i 182 | 1 ⊢ ∪ {𝐴} ∈ {∅, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∅c0 4286 {csn 4590 {cpr 4592 ∪ cuni 4869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-sn 4591 df-pr 4593 df-uni 4870 |
This theorem is referenced by: (None) |
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