Theorem prprc1 2456

Description: A proper class vanishes in an unordered pair.

Assertion

Ref	Expression
prprc1	|- (-. A e. V -> {A, B} = {B})

Proof of Theorem prprc1

Step	Hyp	Ref	Expression
1		snprc 2447	. 2 |- (-. A e. V <-> {A} = (/))
2		uneq1 2180	. . 3 |- ({A} = (/) -> ({A} u. {B}) = ((/) u. {B}))
3		df-pr 2417	. . 3 |- {A, B} = ({A} u. {B})
4		uncom 2179	. . . 4 |- ((/) u. {B}) = ({B} u. (/))
5		un0 2301	. . . 4 |- ({B} u. (/)) = {B}
6	4, 5	eqtr2 1499	. . 3 |- {B} = ((/) u. {B})
7	2, 3, 6	3eqtr4g 1534	. 2 |- ({A} = (/) -> {A, B} = {B})
8	1, 7	sylbi 199	1 |- (-. A e. V -> {A, B} = {B})

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   e. wcel 960  Vcvv 1814   u. cun 2048  (/)c0 2283  {csn 2413  {cpr 2414

This theorem is referenced by:  prprc2 2457  prprc 2458  opprc1 2502  opprc2 2503  prex 2787

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-nul 2284  df-sn 2416  df-pr 2417

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