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Theorem cplem2 9115
Description: Lemma for the Collection Principle cp 9116. (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 cplem2.1 . . 3 𝐴 ∈ V
2 scottex 9110 . . 3 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
31, 2iunex 7483 . 2 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
4 nfiu1 4824 . . . 4 𝑥 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
54nfeq2 2947 . . 3 𝑥 𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
6 ineq2 4072 . . . . 5 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (𝐵𝑦) = (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}))
76neeq1d 3026 . . . 4 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵𝑦) ≠ ∅ ↔ (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅))
87imbi2d 333 . . 3 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
95, 8ralbid 3178 . 2 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ ∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
10 eqid 2778 . . 3 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
11 eqid 2778 . . 3 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
1210, 11cplem1 9114 . 2 𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
133, 9, 12ceqsexv2d 3463 1 𝑦𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wex 1742  wcel 2050  wne 2967  wral 3088  {crab 3092  Vcvv 3415  cin 3830  wss 3831  c0 4180   ciun 4793  cfv 6190  rankcrnk 8988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-reg 8853  ax-inf2 8900
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-iin 4796  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-om 7399  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-r1 8989  df-rank 8990
This theorem is referenced by:  cp  9116
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