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Mirrors > Home > ILE Home > Th. List > prprc1 | GIF version |
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
prprc1 | ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snprc 3588 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | uneq1 3223 | . . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵})) | |
3 | df-pr 3534 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
4 | uncom 3220 | . . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅) | |
5 | un0 3396 | . . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵} | |
6 | 4, 5 | eqtr2i 2161 | . . 3 ⊢ {𝐵} = (∅ ∪ {𝐵}) |
7 | 2, 3, 6 | 3eqtr4g 2197 | . 2 ⊢ ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵}) |
8 | 1, 7 | sylbi 120 | 1 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 ∅c0 3363 {csn 3527 {cpr 3528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-un 3075 df-nul 3364 df-sn 3533 df-pr 3534 |
This theorem is referenced by: prprc2 3632 prprc 3633 |
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