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Mirrors > Home > MPE Home > Th. List > unisn2 | Structured version Visualization version GIF version |
Description: A version of unisn 4921 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.) |
Ref | Expression |
---|---|
unisn2 | ⊢ ∪ {𝐴} ∈ {∅, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisng 4920 | . . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
2 | prid2g 4758 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴}) | |
3 | 1, 2 | eqeltrd 2825 | . 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴}) |
4 | snprc 4714 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
5 | 4 | biimpi 215 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
6 | 5 | unieqd 4913 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
7 | uni0 4930 | . . . 4 ⊢ ∪ ∅ = ∅ | |
8 | 0ex 5298 | . . . . 5 ⊢ ∅ ∈ V | |
9 | 8 | prid1 4759 | . . . 4 ⊢ ∅ ∈ {∅, 𝐴} |
10 | 7, 9 | eqeltri 2821 | . . 3 ⊢ ∪ ∅ ∈ {∅, 𝐴} |
11 | 6, 10 | eqeltrdi 2833 | . 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 1533 ∈ wcel 2098 Vcvv 3466 ∅c0 4315 {csn 4621 {cpr 4623 ∪ cuni 4900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-ext 2695 ax-nul 5297 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-sn 4622 df-pr 4624 df-uni 4901 |
This theorem is referenced by: (None) |
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