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Theorem cplem2 8697
Description: -Lemma for the Collection Principle cp 8698. (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 2621 . . 3 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
2 eqid 2621 . . 3 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
31, 2cplem1 8696 . 2 𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
4 cplem2.1 . . . 4 𝐴 ∈ V
5 scottex 8692 . . . 4 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
64, 5iunex 7093 . . 3 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
7 nfiu1 4516 . . . . 5 𝑥 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
87nfeq2 2776 . . . 4 𝑥 𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
9 ineq2 3786 . . . . . 6 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (𝐵𝑦) = (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}))
109neeq1d 2849 . . . . 5 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵𝑦) ≠ ∅ ↔ (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅))
1110imbi2d 330 . . . 4 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
128, 11ralbid 2977 . . 3 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ ∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
136, 12spcev 3286 . 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 1480  wex 1701  wcel 1987  wne 2790  wral 2907  {crab 2911  Vcvv 3186  cin 3554  wss 3555  c0 3891   ciun 4485  cfv 5847  rankcrnk 8570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-r1 8571  df-rank 8572
This theorem is referenced by:  cp  8698
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