Theorem prprc1 3452

Description: A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
prprc1	⊢ (¬ A ∈ V → {A, B} = {B})

Proof of Theorem prprc1

Step	Hyp	Ref	Expression
1		snprc 3409	. 2 ⊢ (¬ A ∈ V ↔ {A} = ∅)
2		uneq1 3067	. . 3 ⊢ ({A} = ∅ → ({A} ∪ {B}) = (∅ ∪ {B}))
3		df-pr 3357	. . 3 ⊢ {A, B} = ({A} ∪ {B})
4		uncom 3064	. . . 4 ⊢ (∅ ∪ {B}) = ({B} ∪ ∅)
5		un0 3228	. . . 4 ⊢ ({B} ∪ ∅) = {B}
6	4, 5	eqtr2i 2043	. . 3 ⊢ {B} = (∅ ∪ {B})
7	2, 3, 6	3eqtr4g 2079	. 2 ⊢ ({A} = ∅ → {A, B} = {B})
8	1, 7	sylbi 114	1 ⊢ (¬ A ∈ V → {A, B} = {B})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1228   ∈ wcel 1374  Vcvv 2535   ∪ cun 2892  ∅c0 3201  {csn 3350  {cpr 3351

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357

This theorem is referenced by:  prprc2  3453  prprc  3454