Theorem prprc1 4697

Description: A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)

Assertion

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

Proof of Theorem prprc1

Step	Hyp	Ref	Expression
1		snprc 4649	. 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		uneq1 4091	. . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3		df-pr 4558	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		uncom 4088	. . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5		un0 4322	. . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵}
6	4, 5	eqtr2i 2763	. . 3 ⊢ {𝐵} = (∅ ∪ {𝐵})
7	2, 3, 6	3eqtr4g 2799	. 2 ⊢ ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
8	1, 7	sylbi 218	1 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881  ∅c0 4261  {csn 4555  {cpr 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-un 3888  df-nul 4262  df-sn 4556  df-pr 4558

This theorem is referenced by:  prprc2  4698  prprc  4699  prneprprc  4792  prexOLD  5372  prfi  9224  elprchashprn2  14349  prssbd  32618  elsprel  47950