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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 4732 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
| 2 | prprc1 4765 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴}) | |
| 3 | 1, 2 | eqtrid 2789 | 1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 {cpr 4628 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: tpprceq3 4804 elpreqprlem 4866 prex 5437 prfi 9363 indislem 23007 1to2vfriswmgr 30298 indispconn 35239 bj-prmoore 37116 elsprel 47462 |
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