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Theorem cplem2 8914
Description: -Lemma for the Collection Principle cp 8915. (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 2748 . . 3 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
2 eqid 2748 . . 3 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
31, 2cplem1 8913 . 2 𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
4 cplem2.1 . . . 4 𝐴 ∈ V
5 scottex 8909 . . . 4 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
64, 5iunex 7300 . . 3 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
7 nfiu1 4690 . . . . 5 𝑥 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
87nfeq2 2906 . . . 4 𝑥 𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
9 ineq2 3939 . . . . . 6 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (𝐵𝑦) = (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}))
109neeq1d 2979 . . . . 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 3109 . . 3 (𝑦 = 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅) ↔ ∀𝑥𝐴 (𝐵 ≠ ∅ → (𝐵 𝑥𝐴 {𝑧𝐵 ∣ ∀𝑤𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
136, 12spcev 3428 . 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 1620  wex 1841  wcel 2127  wne 2920  wral 3038  {crab 3042  Vcvv 3328  cin 3702  wss 3703  c0 4046   ciun 4660  cfv 6037  rankcrnk 8787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-reg 8650  ax-inf2 8699
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-iin 4663  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-om 7219  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-r1 8788  df-rank 8789
This theorem is referenced by:  cp  8915
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