Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > prprc1 | Structured version Visualization version GIF version |
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.) |
Ref | Expression |
---|---|
prprc1 | ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snprc 4197 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | uneq1 3722 | . . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵})) | |
3 | df-pr 4128 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
4 | uncom 3719 | . . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅) | |
5 | un0 3919 | . . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵} | |
6 | 4, 5 | eqtr2i 2633 | . . 3 ⊢ {𝐵} = (∅ ∪ {𝐵}) |
7 | 2, 3, 6 | 3eqtr4g 2669 | . 2 ⊢ ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵}) |
8 | 1, 7 | sylbi 206 | 1 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∅c0 3874 {csn 4125 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 |
This theorem is referenced by: prprc2 4244 prprc 4245 prex 4836 elprchashprn2 13045 usgraedgprv 25905 |
Copyright terms: Public domain | W3C validator |