Theorem tpnz 3838

Description: A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)

Hypothesis

Ref	Expression
tpnz.1	⊢ A ∈ V

Assertion

Ref	Expression
tpnz	⊢ {A, B, C} ≠ ∅

Proof of Theorem tpnz

Step	Hyp	Ref	Expression
1		tpnz.1	. . 3 ⊢ A ∈ V
2	1	tpid1 3830	. 2 ⊢ A ∈ {A, B, C}
3		ne0i 3557	. 2 ⊢ (A ∈ {A, B, C} → {A, B, C} ≠ ∅)
4	2, 3	ax-mp 5	1 ⊢ {A, B, C} ≠ ∅

Syntax hints:   ∈ wcel 1710   ≠ wne 2517  Vcvv 2860  ∅c0 3551  {ctp 3740

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-3or 935  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  df-tp 3744

This theorem is referenced by: (None)