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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 4211 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | prprc1 4243 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴}) | |
3 | 1, 2 | syl5eq 2656 | 1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {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: tpprceq3 4276 elpreqprlem 4333 prex 4836 indislem 20614 usgraedgprv 25905 1to2vfriswmgra 26533 indispcon 30470 1to2vfriswmgr 41449 |
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