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Mirrors > Home > MPE Home > Th. List > prprc2 | Structured version Visualization version GIF version |
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.) |
Ref | Expression |
---|---|
prprc2 | ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4498 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | prprc1 4531 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴}) | |
3 | 1, 2 | syl5eq 2825 | 1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1601 ∈ wcel 2106 Vcvv 3397 {csn 4397 {cpr 4399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-v 3399 df-dif 3794 df-un 3796 df-nul 4141 df-sn 4398 df-pr 4400 |
This theorem is referenced by: tpprceq3 4566 elpreqprlem 4629 prex 5141 indislem 21212 1to2vfriswmgr 27701 indispconn 31829 elsprel 42406 |
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