Theorem prnzgOLD 4312

Description: Obsolete proof of prnzg 4311 as of 23-Jul-2021. (Contributed by FL, 19-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem prnzgOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 4268	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
2	1	neeq1d 2853	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3		vex 3203	. . 3 ⊢ 𝑥 ∈ V
4	3	prnz 4310	. 2 ⊢ {𝑥, 𝐵} ≠ ∅
5	2, 4	vtoclg 3266	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)