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Theorem fin23lem7 9427
Description: Lemma for isfin2-2 9430. 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 4132 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 difss 3936 . . . . . . . 8 (𝐴𝑦) ⊆ 𝐴
3 elpw2g 5020 . . . . . . . . 9 (𝐴𝑉 → ((𝐴𝑦) ∈ 𝒫 𝐴 ↔ (𝐴𝑦) ⊆ 𝐴))
43ad2antrr 718 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → ((𝐴𝑦) ∈ 𝒫 𝐴 ↔ (𝐴𝑦) ⊆ 𝐴))
52, 4mpbiri 250 . . . . . . 7 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ 𝒫 𝐴)
6 simpr 478 . . . . . . . . . . 11 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → 𝐵 ⊆ 𝒫 𝐴)
76sselda 3799 . . . . . . . . . 10 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ 𝒫 𝐴)
87elpwid 4362 . . . . . . . . 9 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦𝐴)
9 dfss4 4060 . . . . . . . . 9 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
108, 9sylib 210 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴 ∖ (𝐴𝑦)) = 𝑦)
11 simpr 478 . . . . . . . 8 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → 𝑦𝐵)
1210, 11eqeltrd 2879 . . . . . . 7 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵)
13 difeq2 3921 . . . . . . . . 9 (𝑥 = (𝐴𝑦) → (𝐴𝑥) = (𝐴 ∖ (𝐴𝑦)))
1413eleq1d 2864 . . . . . . . 8 (𝑥 = (𝐴𝑦) → ((𝐴𝑥) ∈ 𝐵 ↔ (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵))
1514rspcev 3498 . . . . . . 7 (((𝐴𝑦) ∈ 𝒫 𝐴 ∧ (𝐴 ∖ (𝐴𝑦)) ∈ 𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
165, 12, 15syl2anc 580 . . . . . 6 (((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
1716ex 402 . . . . 5 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
1817exlimdv 2029 . . . 4 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (∃𝑦 𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
191, 18syl5bi 234 . . 3 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴) → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵))
20193impia 1146 . 2 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
21 rabn0 4159 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ 𝐵)
2220, 21sylibr 226 1 ((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  wne 2972  wrex 3091  {crab 3094  cdif 3767  wss 3770  c0 4116  𝒫 cpw 4350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-in 3777  df-ss 3784  df-nul 4117  df-pw 4352
This theorem is referenced by:  fin2i2  9429  isfin2-2  9430
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