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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 4722 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | uneq1 4171 | . . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵})) | |
3 | df-pr 4634 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
4 | uncom 4168 | . . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅) | |
5 | un0 4400 | . . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵} | |
6 | 4, 5 | eqtr2i 2764 | . . 3 ⊢ {𝐵} = (∅ ∪ {𝐵}) |
7 | 2, 3, 6 | 3eqtr4g 2800 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∅c0 4339 {csn 4631 {cpr 4633 |
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-ext 2706 |
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-v 3480 df-dif 3966 df-un 3968 df-nul 4340 df-sn 4632 df-pr 4634 |
This theorem is referenced by: prprc2 4771 prprc 4772 prneprprc 4866 prex 5443 prfi 9361 elprchashprn2 14432 elsprel 47400 |
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