Theorem prprc1 4729

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 4681	. 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		uneq1 4124	. . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3		df-pr 4592	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		uncom 4121	. . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5		un0 4357	. . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵}
6	4, 5	eqtr2i 2753	. . 3 ⊢ {𝐵} = (∅ ∪ {𝐵})
7	2, 3, 6	3eqtr4g 2789	. 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 2109  Vcvv 3447   ∪ cun 3912  ∅c0 4296  {csn 4589  {cpr 4591

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-dif 3917  df-un 3919  df-nul 4297  df-sn 4590  df-pr 4592

This theorem is referenced by:  prprc2  4730  prprc  4731  prneprprc  4825  prex  5392  prfi  9274  elprchashprn2  14361  prssbd  32459  elsprel  47476