|   | 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 4717 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 2 | uneq1 4161 | . . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵})) | |
| 3 | df-pr 4629 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 4 | uncom 4158 | . . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅) | |
| 5 | un0 4394 | . . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵} | |
| 6 | 4, 5 | eqtr2i 2766 | . . 3 ⊢ {𝐵} = (∅ ∪ {𝐵}) | 
| 7 | 2, 3, 6 | 3eqtr4g 2802 | . 2 ⊢ ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵}) | 
| 8 | 1, 7 | sylbi 217 | 1 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∅c0 4333 {csn 4626 {cpr 4628 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: prprc2 4766 prprc 4767 prneprprc 4861 prex 5437 prfi 9363 elprchashprn2 14435 elsprel 47462 | 
| Copyright terms: Public domain | W3C validator |