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Theorem cplem2 8636
Description: -Lemma for the Collection Principle cp 8637. (Contributed by NM, 17-Oct-2003.)
Hypothesis
Ref Expression
cplem2.1 𝐴 ∈ V
Assertion
Ref Expression
cplem2 𝑦𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem cplem2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
2 eqid 2610 . . 3 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
31, 2cplem1 8635 . 2 𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
4 cplem2.1 . . . 4 𝐴 ∈ V
5 scottex 8631 . . . 4 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
64, 5iunex 7039 . . 3 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
7 nfiu1 4486 . . . . 5 𝑥 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
87nfeq2 2766 . . . 4 𝑥 𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
9 ineq2 3770 . . . . . 6 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (𝐵𝑦) = (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}))
109neeq1d 2841 . . . . 5 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵𝑦) ≠ ∅ ↔ (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅))
1110imbi2d 329 . . . 4 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
128, 11ralbid 2966 . . 3 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ ∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
136, 12spcev 3273 . 2 (∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅) → ∃𝑦𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅))
143, 13ax-mp 5 1 𝑦𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wex 1695  wcel 1977  wne 2780  wral 2896  {crab 2900  Vcvv 3173  cin 3539  wss 3540  c0 3874   ciun 4455  cfv 5804  rankcrnk 8509
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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511
This theorem is referenced by:  cp  8637
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