Theorem prnzg 3837

Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
prnzg	⊢ (A ∈ V → {A, B} ≠ ∅)

Proof of Theorem prnzg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 3800	. . 3 ⊢ (x = A → {x, B} = {A, B})
2	1	neeq1d 2530	. 2 ⊢ (x = A → ({x, B} ≠ ∅ ↔ {A, B} ≠ ∅))
3		vex 2863	. . 3 ⊢ x ∈ V
4	3	prnz 3836	. 2 ⊢ {x, B} ≠ ∅
5	2, 4	vtoclg 2915	1 ⊢ (A ∈ V → {A, B} ≠ ∅)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∅c0 3551  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-nul 3552  df-sn 3742  df-pr 3743

This theorem is referenced by: (None)