Theorem prprc1 4741

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 4693	. 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		uneq1 4136	. . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3		df-pr 4604	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		uncom 4133	. . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5		un0 4369	. . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵}
6	4, 5	eqtr2i 2759	. . 3 ⊢ {𝐵} = (∅ ∪ {𝐵})
7	2, 3, 6	3eqtr4g 2795	. 2 ⊢ ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
8	1, 7	sylbi 217	1 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924  ∅c0 4308  {csn 4601  {cpr 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-dif 3929  df-un 3931  df-nul 4309  df-sn 4602  df-pr 4604

This theorem is referenced by:  prprc2  4742  prprc  4743  prneprprc  4837  prex  5407  prfi  9335  elprchashprn2  14414  prssbd  32511  elsprel  47489