Theorem bj-pr0moore 38112

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 5260	. 2 ⊢ ∅ ∈ V
2		0ss 4349	. 2 ⊢ ∅ ⊆ 𝐴
3		bj-prmoore 38111	. 2 ⊢ ((∅ ∈ V ∧ ∅ ⊆ 𝐴) → {∅, 𝐴} ∈ Moore)
4	1, 2, 3	mp2an 705	1 ⊢ {∅, 𝐴} ∈ Moore

This proof depends on syntax axioms:   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  {cpr 4585  Moorecmoore 38099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-pw 4558  df-sn 4584  df-pr 4586  df-uni 4867  df-int 4907  df-bj-moore 38100

This theorem is used by: (None)