Theorem prprc1 3469

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 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})

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-bnd 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