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Mirrors > Home > ILE Home > Th. List > prprc2 | 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 3522 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | prprc1 3554 | . 2 ⊢ (¬ 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴}) | |
3 | 1, 2 | syl5eq 2133 | 1 ⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1290 ∈ wcel 1439 Vcvv 2620 {csn 3450 {cpr 3451 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-dif 3002 df-un 3004 df-nul 3288 df-sn 3456 df-pr 3457 |
This theorem is referenced by: (None) |
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