Theorem prnzg 3833

Description: A pair containing a set is not empty. It is also inhabited (see prmg 3830). (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 3784	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
2	1	neeq1d 2438	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3		vex 2824	. . 3 ⊢ 𝑥 ∈ V
4	3	prnz 3831	. 2 ⊢ {𝑥, 𝐵} ≠ ∅
5	2, 4	vtoclg 2883	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∅c0 3520  {cpr 3706

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3711  df-pr 3712

This theorem is referenced by:  0nelop  4383