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Theorem bj-ismooredr2 37111
Description: Sufficient condition to be a Moore collection (variant of bj-ismooredr 37110 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 467 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)
3 intssuni2 4973 . . . . . . 7 ((𝑥𝐴𝑥 ≠ ∅) → 𝑥 𝐴)
4 dfss 3970 . . . . . . . 8 ( 𝑥 𝐴 𝑥 = ( 𝑥 𝐴))
5 incom 4209 . . . . . . . . . . 11 ( 𝑥 𝐴) = ( 𝐴 𝑥)
65eqeq2i 2750 . . . . . . . . . 10 ( 𝑥 = ( 𝑥 𝐴) ↔ 𝑥 = ( 𝐴 𝑥))
7 eleq1 2829 . . . . . . . . . 10 ( 𝑥 = ( 𝐴 𝑥) → ( 𝑥𝐴 ↔ ( 𝐴 𝑥) ∈ 𝐴))
86, 7sylbi 217 . . . . . . . . 9 ( 𝑥 = ( 𝑥 𝐴) → ( 𝑥𝐴 ↔ ( 𝐴 𝑥) ∈ 𝐴))
98biimpd 229 . . . . . . . 8 ( 𝑥 = ( 𝑥 𝐴) → ( 𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
104, 9sylbi 217 . . . . . . 7 ( 𝑥 𝐴 → ( 𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
113, 10syl 17 . . . . . 6 ((𝑥𝐴𝑥 ≠ ∅) → ( 𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
1211adantll 714 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑥 ≠ ∅) → ( 𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
132, 12mpd 15 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑥 ≠ ∅) → ( 𝐴 𝑥) ∈ 𝐴)
1413ex 412 . . 3 ((𝜑𝑥𝐴) → (𝑥 ≠ ∅ → ( 𝐴 𝑥) ∈ 𝐴))
15 nne 2944 . . . . 5 𝑥 ≠ ∅ ↔ 𝑥 = ∅)
16 bj-ismooredr2.1 . . . . . 6 (𝜑 𝐴𝐴)
17 rint0 4988 . . . . . 6 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
18 eleq1a 2836 . . . . . 6 ( 𝐴𝐴 → (( 𝐴 𝑥) = 𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
1916, 17, 18syl2im 40 . . . . 5 (𝜑 → (𝑥 = ∅ → ( 𝐴 𝑥) ∈ 𝐴))
2015, 19biimtrid 242 . . . 4 (𝜑 → (¬ 𝑥 ≠ ∅ → ( 𝐴 𝑥) ∈ 𝐴))
2120adantr 480 . . 3 ((𝜑𝑥𝐴) → (¬ 𝑥 ≠ ∅ → ( 𝐴 𝑥) ∈ 𝐴))
2214, 21pm2.61d 179 . 2 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
2322bj-ismooredr 37110 1 (𝜑𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  cin 3950  wss 3951  c0 4333   cuni 4907   cint 4946  Moorecmoore 37104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-uni 4908  df-int 4947  df-bj-moore 37105
This theorem is referenced by:  bj-snmoore  37114  bj-prmoore  37116
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