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Theorem fin23lem7 10311
Description: Lemma for isfin2-2 10314. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
fin23lem7 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem fin23lem7
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 n0 4347 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 difss 4132 . . . . . . . 8 (𝐴𝑦) ⊆ 𝐴
3 elpw2g 5345 . . . . . . . . 9 (𝐴𝑉 → ((𝐴𝑦) ∈ 𝒫 𝐴 ↔ (𝐴𝑦) ⊆ 𝐴))
43ad2antrr 725 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → ((𝐴𝑦) ∈ 𝒫 𝐴 ↔ (𝐴𝑦) ⊆ 𝐴))
52, 4mpbiri 258 . . . . . . 7 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ 𝒫 𝐴)
6 simpr 486 . . . . . . . . . . 11 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → 𝐵 ⊆ 𝒫 𝐴)
76sselda 3983 . . . . . . . . . 10 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ 𝒫 𝐴)
87elpwid 4612 . . . . . . . . 9 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦𝐴)
9 dfss4 4259 . . . . . . . . 9 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
108, 9sylib 217 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴 ∖ (𝐴𝑦)) = 𝑦)
11 simpr 486 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦𝐵)
1210, 11eqeltrd 2834 . . . . . . 7 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵)
13 difeq2 4117 . . . . . . . . 9 (𝑥 = (𝐴𝑦) → (𝐴𝑥) = (𝐴 ∖ (𝐴𝑦)))
1413eleq1d 2819 . . . . . . . 8 (𝑥 = (𝐴𝑦) → ((𝐴𝑥) ∈ 𝐵 ↔ (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵))
1514rspcev 3613 . . . . . . 7 (((𝐴𝑦) ∈ 𝒫 𝐴 ∧ (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
165, 12, 15syl2anc 585 . . . . . 6 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
1716ex 414 . . . . 5 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
1817exlimdv 1937 . . . 4 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (∃𝑦 𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
191, 18biimtrid 241 . . 3 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
20193impia 1118 . 2 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
21 rabn0 4386 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
2220, 21sylibr 233 1 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wne 2941  wrex 3071  {crab 3433  cdif 3946  wss 3949  c0 4323  𝒫 cpw 4603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605
This theorem is referenced by:  fin2i2  10313  isfin2-2  10314
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