Theorem bj-ismooredr2 34404

Description: Sufficient condition to be a Moore collection (variant of bj-ismooredr 34403 singling out the empty intersection). Note that there is no sethood hypothesis on 𝐴: it is a consequence of the first hypothesis. (Contributed by BJ, 9-Dec-2021.)

Hypotheses

Ref	Expression
bj-ismooredr2.1	⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)
bj-ismooredr2.2	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
bj-ismooredr2	⊢ (𝜑 → 𝐴 ∈ Moore)

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

Proof of Theorem bj-ismooredr2

Step	Hyp	Ref	Expression
1		bj-ismooredr2.2	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ 𝐴)
2	1	anassrs 470	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐴)
3		intssuni2 4903	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝐴)
4		dfss 3955	. . . . . . . 8 ⊢ (∩ 𝑥 ⊆ ∪ 𝐴 ↔ ∩ 𝑥 = (∩ 𝑥 ∩ ∪ 𝐴))
5		incom 4180	. . . . . . . . . . 11 ⊢ (∩ 𝑥 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ ∩ 𝑥)
6	5	eqeq2i 2836	. . . . . . . . . 10 ⊢ (∩ 𝑥 = (∩ 𝑥 ∩ ∪ 𝐴) ↔ ∩ 𝑥 = (∪ 𝐴 ∩ ∩ 𝑥))
7		eleq1 2902	. . . . . . . . . 10 ⊢ (∩ 𝑥 = (∪ 𝐴 ∩ ∩ 𝑥) → (∩ 𝑥 ∈ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
8	6, 7	sylbi 219	. . . . . . . . 9 ⊢ (∩ 𝑥 = (∩ 𝑥 ∩ ∪ 𝐴) → (∩ 𝑥 ∈ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
9	8	biimpd 231	. . . . . . . 8 ⊢ (∩ 𝑥 = (∩ 𝑥 ∩ ∪ 𝐴) → (∩ 𝑥 ∈ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
10	4, 9	sylbi 219	. . . . . . 7 ⊢ (∩ 𝑥 ⊆ ∪ 𝐴 → (∩ 𝑥 ∈ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
11	3, 10	syl 17	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → (∩ 𝑥 ∈ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
12	11	adantll 712	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐴) ∧ 𝑥 ≠ ∅) → (∩ 𝑥 ∈ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
13	2, 12	mpd 15	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐴) ∧ 𝑥 ≠ ∅) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
14	13	ex 415	. . 3 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (𝑥 ≠ ∅ → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
15		nne 3022	. . . . 5 ⊢ (¬ 𝑥 ≠ ∅ ↔ 𝑥 = ∅)
16		bj-ismooredr2.1	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)
17		rint0 4918	. . . . . 6 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴)
18		eleq1a 2910	. . . . . 6 ⊢ (∪ 𝐴 ∈ 𝐴 → ((∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
19	16, 17, 18	syl2im 40	. . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
20	15, 19	syl5bi 244	. . . 4 ⊢ (𝜑 → (¬ 𝑥 ≠ ∅ → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
21	20	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (¬ 𝑥 ≠ ∅ → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
22	14, 21	pm2.61d 181	. 2 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
23	22	bj-ismooredr 34403	1 ⊢ (𝜑 → 𝐴 ∈ Moore)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  ∪ cuni 4840  ∩ cint 4878  Moorecmoore 34397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-in 3945  df-ss 3954  df-nul 4294  df-pw 4543  df-uni 4841  df-int 4879  df-bj-moore 34398

This theorem is referenced by:  bj-snmoore  34407  bj-prmoore  34409