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Theorem cplem2 9307
Description: Lemma for the Collection Principle cp 9308. (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 9302 . . 3 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
31, 2iunex 7655 . 2 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
4 nfiu1 4928 . . . 4 𝑥 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
54nfeq2 2996 . . 3 𝑥 𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
6 ineq2 4157 . . . . 5 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (𝐵𝑦) = (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}))
76neeq1d 3070 . . . 4 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵𝑦) ≠ ∅ ↔ (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅))
87imbi2d 344 . . 3 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
95, 8ralbid 3220 . 2 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ ∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
10 eqid 2822 . . 3 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
11 eqid 2822 . . 3 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
1210, 11cplem1 9306 . 2 𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
133, 9, 12ceqsexv2d 3517 1 𝑦𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wex 1781  wcel 2114  wne 3011  wral 3130  {crab 3134  Vcvv 3469  cin 3907  wss 3908  c0 4265   ciun 4894  cfv 6334  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181  df-rank 9182
This theorem is referenced by:  cp  9308
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