Theorem cplem2 9928

Description: Lemma for the Collection Principle cp 9929. (Contributed by NM, 17-Oct-2003.)

Hypothesis

Ref	Expression
cplem2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cplem2	⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)

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.

Step	Hyp	Ref	Expression
1		cplem2.1	. . 3 ⊢ 𝐴 ∈ V
2		scottex 9923	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
3	1, 2	iunex 7992	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
4		nfiu1 5032	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
5	4	nfeq2 2921	. . 3 ⊢ Ⅎ𝑥 𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
6		ineq2 4222	. . . . 5 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (𝐵 ∩ 𝑦) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}))
7	6	neeq1d 2998	. . . 4 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵 ∩ 𝑦) ≠ ∅ ↔ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅))
8	7	imbi2d 340	. . 3 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → ((𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅) ↔ (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
9	5, 8	ralbid 3271	. 2 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} → (∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅) ↔ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)))
10		eqid 2735	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
11		eqid 2735	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
12	10, 11	cplem1 9927	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
13	3, 9, 12	ceqsexv2d 3533	1 ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)

Syntax hints:   → wi 4   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  {crab 3433  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  ∪ ciun 4996  ‘cfv 6563  rankcrnk 9801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803

This theorem is referenced by:  cp  9929