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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 4679 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | uneq1 4117 | . . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵})) | |
3 | df-pr 4590 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
4 | uncom 4114 | . . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅) | |
5 | un0 4351 | . . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵} | |
6 | 4, 5 | eqtr2i 2762 | . . 3 ⊢ {𝐵} = (∅ ∪ {𝐵}) |
7 | 2, 3, 6 | 3eqtr4g 2798 | . 2 ⊢ ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵}) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∪ cun 3909 ∅c0 4283 {csn 4587 {cpr 4589 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-dif 3914 df-un 3916 df-nul 4284 df-sn 4588 df-pr 4590 |
This theorem is referenced by: prprc2 4728 prprc 4729 prneprprc 4819 prex 5390 elprchashprn2 14302 elsprel 45753 |
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