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Mirrors > Home > MPE Home > Th. List > unisn2 | Structured version Visualization version GIF version |
Description: A version of unisn 4861 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.) |
Ref | Expression |
---|---|
unisn2 | ⊢ ∪ {𝐴} ∈ {∅, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisng 4860 | . . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
2 | prid2g 4697 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴}) | |
3 | 1, 2 | eqeltrd 2839 | . 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴}) |
4 | snprc 4653 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
5 | 4 | biimpi 215 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
6 | 5 | unieqd 4853 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
7 | uni0 4869 | . . . 4 ⊢ ∪ ∅ = ∅ | |
8 | 0ex 5231 | . . . . 5 ⊢ ∅ ∈ V | |
9 | 8 | prid1 4698 | . . . 4 ⊢ ∅ ∈ {∅, 𝐴} |
10 | 7, 9 | eqeltri 2835 | . . 3 ⊢ ∪ ∅ ∈ {∅, 𝐴} |
11 | 6, 10 | eqeltrdi 2847 | . 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 1539 ∈ wcel 2106 Vcvv 3432 ∅c0 4256 {csn 4561 {cpr 4563 ∪ cuni 4839 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-sn 4562 df-pr 4564 df-uni 4840 |
This theorem is referenced by: (None) |
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