Theorem cplem2OLD 9895

Description: Obsolete version of cplem2 9894 as of 19-Jul-2026. (Contributed by NM, 17-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cplem2OLD.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cplem2OLD	⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem cplem2OLD

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cplem2OLD.1	. . 3 ⊢ 𝐴 ∈ V
2		scottexOLD 9876	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
3	1, 2	iunex 7966	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
4		nfiu1 4986	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
5	4	nfeq2 2939	. . 3 ⊢ Ⅎ𝑥 𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
6		ineq2 4160	. . . . 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 3014	. . . 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 343	. . 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 3275	. 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 2760	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
11		eqid 2760	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
12	10, 11	cplem1OLD 9893	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
13	3, 9, 12	ceqsexv2d 3499	1 ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  {crab 3412  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ∪ ciun 4951  ‘cfv 6533  rankcrnk 9748

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-reg 9567  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-r1 9749  df-rank 9750

This theorem is used by: (None)