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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 4660 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | prprc1 4693 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴}) | |
3 | 1, 2 | syl5eq 2865 | 1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 {cpr 4559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-un 3938 df-nul 4289 df-sn 4558 df-pr 4560 |
This theorem is referenced by: tpprceq3 4729 elpreqprlem 4788 prex 5323 indislem 21536 1to2vfriswmgr 27985 indispconn 32378 elsprel 43514 |
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