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Theorem fin23lem7 9176
Description: Lemma for isfin2-2 9179. 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 3964 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 difss 3770 . . . . . . . 8 (𝐴𝑦) ⊆ 𝐴
3 elpw2g 4857 . . . . . . . . 9 (𝐴𝑉 → ((𝐴𝑦) ∈ 𝒫 𝐴 ↔ (𝐴𝑦) ⊆ 𝐴))
43ad2antrr 762 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → ((𝐴𝑦) ∈ 𝒫 𝐴 ↔ (𝐴𝑦) ⊆ 𝐴))
52, 4mpbiri 248 . . . . . . 7 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ 𝒫 𝐴)
6 simpr 476 . . . . . . . . . . 11 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → 𝐵 ⊆ 𝒫 𝐴)
76sselda 3636 . . . . . . . . . 10 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ 𝒫 𝐴)
87elpwid 4203 . . . . . . . . 9 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦𝐴)
9 dfss4 3891 . . . . . . . . 9 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
108, 9sylib 208 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴 ∖ (𝐴𝑦)) = 𝑦)
11 simpr 476 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦𝐵)
1210, 11eqeltrd 2730 . . . . . . 7 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵)
13 difeq2 3755 . . . . . . . . 9 (𝑥 = (𝐴𝑦) → (𝐴𝑥) = (𝐴 ∖ (𝐴𝑦)))
1413eleq1d 2715 . . . . . . . 8 (𝑥 = (𝐴𝑦) → ((𝐴𝑥) ∈ 𝐵 ↔ (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵))
1514rspcev 3340 . . . . . . 7 (((𝐴𝑦) ∈ 𝒫 𝐴 ∧ (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
165, 12, 15syl2anc 694 . . . . . 6 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
1716ex 449 . . . . 5 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
1817exlimdv 1901 . . . 4 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (∃𝑦 𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
191, 18syl5bi 232 . . 3 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
20193impia 1280 . 2 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
21 rabn0 3991 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
2220, 21sylibr 224 1 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wrex 2942  {crab 2945  cdif 3604  wss 3607  c0 3948  𝒫 cpw 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193
This theorem is referenced by:  fin2i2  9178  isfin2-2  9179
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