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Theorem bj-ismooredr2 34839
 Description: Sufficient condition to be a Moore collection (variant of bj-ismooredr 34838 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 𝐴𝐴)
bj-ismooredr2.2 ((𝜑 ∧ (𝑥𝐴𝑥 ≠ ∅)) → 𝑥𝐴)
Assertion
Ref Expression
bj-ismooredr2 (𝜑𝐴Moore)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

Proof of Theorem bj-ismooredr2
StepHypRef Expression
1 bj-ismooredr2.2 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑥 ≠ ∅)) → 𝑥𝐴)
21anassrs 471 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)
3 intssuni2 4866 . . . . . . 7 ((𝑥𝐴𝑥 ≠ ∅) → 𝑥 𝐴)
4 dfss 3878 . . . . . . . 8 ( 𝑥 𝐴 𝑥 = ( 𝑥 𝐴))
5 incom 4108 . . . . . . . . . . 11 ( 𝑥 𝐴) = ( 𝐴 𝑥)
65eqeq2i 2771 . . . . . . . . . 10 ( 𝑥 = ( 𝑥 𝐴) ↔ 𝑥 = ( 𝐴 𝑥))
7 eleq1 2839 . . . . . . . . . 10 ( 𝑥 = ( 𝐴 𝑥) → ( 𝑥𝐴 ↔ ( 𝐴 𝑥) ∈ 𝐴))
86, 7sylbi 220 . . . . . . . . 9 ( 𝑥 = ( 𝑥 𝐴) → ( 𝑥𝐴 ↔ ( 𝐴 𝑥) ∈ 𝐴))
98biimpd 232 . . . . . . . 8 ( 𝑥 = ( 𝑥 𝐴) → ( 𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
104, 9sylbi 220 . . . . . . 7 ( 𝑥 𝐴 → ( 𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
113, 10syl 17 . . . . . 6 ((𝑥𝐴𝑥 ≠ ∅) → ( 𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
1211adantll 713 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑥 ≠ ∅) → ( 𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
132, 12mpd 15 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑥 ≠ ∅) → ( 𝐴 𝑥) ∈ 𝐴)
1413ex 416 . . 3 ((𝜑𝑥𝐴) → (𝑥 ≠ ∅ → ( 𝐴 𝑥) ∈ 𝐴))
15 nne 2955 . . . . 5 𝑥 ≠ ∅ ↔ 𝑥 = ∅)
16 bj-ismooredr2.1 . . . . . 6 (𝜑 𝐴𝐴)
17 rint0 4883 . . . . . 6 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
18 eleq1a 2847 . . . . . 6 ( 𝐴𝐴 → (( 𝐴 𝑥) = 𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
1916, 17, 18syl2im 40 . . . . 5 (𝜑 → (𝑥 = ∅ → ( 𝐴 𝑥) ∈ 𝐴))
2015, 19syl5bi 245 . . . 4 (𝜑 → (¬ 𝑥 ≠ ∅ → ( 𝐴 𝑥) ∈ 𝐴))
2120adantr 484 . . 3 ((𝜑𝑥𝐴) → (¬ 𝑥 ≠ ∅ → ( 𝐴 𝑥) ∈ 𝐴))
2214, 21pm2.61d 182 . 2 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
2322bj-ismooredr 34838 1 (𝜑𝐴Moore)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ∪ cuni 4801  ∩ cint 4841  Moorecmoore 34832 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-in 3867  df-ss 3877  df-nul 4228  df-pw 4499  df-uni 4802  df-int 4842  df-bj-moore 34833 This theorem is referenced by:  bj-snmoore  34842  bj-prmoore  34844
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