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Theorem fin23lem7 9021
 Description: Lemma for isfin2-2 9024. 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 3890 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 difss 3699 . . . . . . . 8 (𝐴𝑦) ⊆ 𝐴
3 elpw2g 4754 . . . . . . . . 9 (𝐴𝑉 → ((𝐴𝑦) ∈ 𝒫 𝐴 ↔ (𝐴𝑦) ⊆ 𝐴))
43ad2antrr 758 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → ((𝐴𝑦) ∈ 𝒫 𝐴 ↔ (𝐴𝑦) ⊆ 𝐴))
52, 4mpbiri 247 . . . . . . 7 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ 𝒫 𝐴)
6 simpr 476 . . . . . . . . . . 11 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → 𝐵 ⊆ 𝒫 𝐴)
76sselda 3568 . . . . . . . . . 10 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ 𝒫 𝐴)
87elpwid 4118 . . . . . . . . 9 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦𝐴)
9 dfss4 3820 . . . . . . . . 9 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
108, 9sylib 207 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴 ∖ (𝐴𝑦)) = 𝑦)
11 simpr 476 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦𝐵)
1210, 11eqeltrd 2688 . . . . . . 7 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵)
13 difeq2 3684 . . . . . . . . 9 (𝑥 = (𝐴𝑦) → (𝐴𝑥) = (𝐴 ∖ (𝐴𝑦)))
1413eleq1d 2672 . . . . . . . 8 (𝑥 = (𝐴𝑦) → ((𝐴𝑥) ∈ 𝐵 ↔ (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵))
1514rspcev 3282 . . . . . . 7 (((𝐴𝑦) ∈ 𝒫 𝐴 ∧ (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
165, 12, 15syl2anc 691 . . . . . 6 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
1716ex 449 . . . . 5 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
1817exlimdv 1848 . . . 4 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (∃𝑦 𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
191, 18syl5bi 231 . . 3 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
20193impia 1253 . 2 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
21 rabn0 3912 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
2220, 21sylibr 223 1 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110 This theorem is referenced by:  fin2i2  9023  isfin2-2  9024
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