![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > unisn2 | Structured version Visualization version GIF version |
Description: A version of unisn 4931 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.) |
Ref | Expression |
---|---|
unisn2 | ⊢ ∪ {𝐴} ∈ {∅, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisng 4930 | . . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴) | |
2 | prid2g 4766 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴}) | |
3 | 1, 2 | eqeltrd 2839 | . 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴}) |
4 | snprc 4722 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
5 | 4 | biimpi 216 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
6 | 5 | unieqd 4925 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅) |
7 | uni0 4940 | . . . 4 ⊢ ∪ ∅ = ∅ | |
8 | 0ex 5313 | . . . . 5 ⊢ ∅ ∈ V | |
9 | 8 | prid1 4767 | . . . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 {cpr 4633 ∪ cuni 4912 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-uni 4913 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |