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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 4671 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | prprc1 4704 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴}) | |
3 | 1, 2 | syl5eq 2871 | 1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 {cpr 4572 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-dif 3942 df-un 3944 df-nul 4295 df-sn 4571 df-pr 4573 |
This theorem is referenced by: tpprceq3 4740 elpreqprlem 4799 prex 5336 indislem 21611 1to2vfriswmgr 28061 indispconn 32485 bj-prmoore 34411 elsprel 43644 |
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