Theorem bj-pr0moore 38228

Description: A pair containing the empty set is a Moore collection. (Contributed by BJ, 4-Oct-2026.)

Assertion

Ref	Expression
bj-pr0moore	⊢ {∅, 𝐴} ∈ Moore

Proof of Theorem bj-pr0moore

Step	Hyp	Ref	Expression
1		0ex 5264	. 2 ⊢ ∅ ∈ V
2		0ss 4353	. 2 ⊢ ∅ ⊆ 𝐴
3		bj-prmoore 38227	. 2 ⊢ ((∅ ∈ V ∧ ∅ ⊆ 𝐴) → {∅, 𝐴} ∈ Moore)
4	1, 2, 3	mp2an 705	1 ⊢ {∅, 𝐴} ∈ Moore

This proof depends on syntax axioms:   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {cpr 4590  Moorecmoore 38215

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-pw 4562  df-sn 4589  df-pr 4591  df-uni 4871  df-int 4911  df-bj-moore 38216

This theorem is used by: (None)