Theorem prprc1 4724

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 4676	. 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		uneq1 4114	. . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3		df-pr 4587	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		uncom 4111	. . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5		un0 4348	. . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵}
6	4, 5	eqtr2i 2765	. . 3 ⊢ {𝐵} = (∅ ∪ {𝐵})
7	2, 3, 6	3eqtr4g 2801	. 2 ⊢ ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
8	1, 7	sylbi 216	1 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3443   ∪ cun 3906  ∅c0 4280  {csn 4584  {cpr 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3445  df-dif 3911  df-un 3913  df-nul 4281  df-sn 4585  df-pr 4587

This theorem is referenced by:  prprc2  4725  prprc  4726  prneprprc  4816  prex  5387  elprchashprn2  14250  elsprel  45562