Theorem prnzg 3817

Description: A pair containing a set is not empty. It is also inhabited (see prmg 3814). (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 3768	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
2	1	neeq1d 2430	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3		vex 2816	. . 3 ⊢ 𝑥 ∈ V
4	3	prnz 3815	. 2 ⊢ {𝑥, 𝐵} ≠ ∅
5	2, 4	vtoclg 2875	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∅c0 3508  {cpr 3690

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-v 2815  df-dif 3213  df-un 3215  df-nul 3509  df-sn 3695  df-pr 3696

This theorem is referenced by:  0nelop  4364