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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 3426 | . 2 ⊢ (¬ A ∈ V ↔ {A} = ∅) | |
2 | uneq1 3084 | . . 3 ⊢ ({A} = ∅ → ({A} ∪ {B}) = (∅ ∪ {B})) | |
3 | df-pr 3374 | . . 3 ⊢ {A, B} = ({A} ∪ {B}) | |
4 | uncom 3081 | . . . 4 ⊢ (∅ ∪ {B}) = ({B} ∪ ∅) | |
5 | un0 3245 | . . . 4 ⊢ ({B} ∪ ∅) = {B} | |
6 | 4, 5 | eqtr2i 2058 | . . 3 ⊢ {B} = (∅ ∪ {B}) |
7 | 2, 3, 6 | 3eqtr4g 2094 | . 2 ⊢ ({A} = ∅ → {A, B} = {B}) |
8 | 1, 7 | sylbi 114 | 1 ⊢ (¬ A ∈ V → {A, B} = {B}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ∪ cun 2909 ∅c0 3218 {csn 3367 {cpr 3368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-dif 2914 df-un 2916 df-nul 3219 df-sn 3373 df-pr 3374 |
This theorem is referenced by: prprc2 3470 prprc 3471 |
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