Theorem prnzg 3768

Description: A pair containing a set is not empty. It is also inhabited (see prmg 3765). (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 3720	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
2	1	neeq1d 2396	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3		vex 2779	. . 3 ⊢ 𝑥 ∈ V
4	3	prnz 3766	. 2 ⊢ {𝑥, 𝐵} ≠ ∅
5	2, 4	vtoclg 2838	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2178   ≠ wne 2378  ∅c0 3468  {cpr 3644

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778  df-dif 3176  df-un 3178  df-nul 3469  df-sn 3649  df-pr 3650

This theorem is referenced by:  0nelop  4310