Theorem prprc1 3780

Description: A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem prprc1

Step	Hyp	Ref	Expression
1		snprc 3734	. 2 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		uneq1 3354	. . 3 ⊢ ({𝐴} = ∅ → ({𝐴} ∪ {𝐵}) = (∅ ∪ {𝐵}))
3		df-pr 3676	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4		uncom 3351	. . . 4 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
5		un0 3528	. . . 4 ⊢ ({𝐵} ∪ ∅) = {𝐵}
6	4, 5	eqtr2i 2253	. . 3 ⊢ {𝐵} = (∅ ∪ {𝐵})
7	2, 3, 6	3eqtr4g 2289	. 2 ⊢ ({𝐴} = ∅ → {𝐴, 𝐵} = {𝐵})
8	1, 7	sylbi 121	1 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∪ cun 3198  ∅c0 3494  {csn 3669  {cpr 3670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-sn 3675  df-pr 3676

This theorem is referenced by:  prprc2  3781  prprc  3782