Theorem cplem2 9303

Description: Lemma for the Collection Principle cp 9304. (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 9298	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
3	1, 2	iunex 7651	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
4		nfiu1 4915	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
5	4	nfeq2 2972	. . 3 ⊢ Ⅎ𝑥 𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
6		ineq2 4133	. . . . 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 3046	. . . 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 344	. . 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 3195	. 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 2798	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
11		eqid 2798	. . 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 9302	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
13	3, 9, 12	ceqsexv2d 3490	1 ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)

Syntax hints:   → wi 4   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ∪ ciun 4881  ‘cfv 6324  rankcrnk 9176

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040  ax-inf2 9088

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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-r1 9177  df-rank 9178

This theorem is referenced by:  cp  9304