Theorem cplem2OLD 9966

Description: Obsolete version of cplem2 9965 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 9947	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
3	1, 2	iunex 7990	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} ∈ V
4		nfiu1 4989	. . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
5	4	nfeq2 2943	. . 3 ⊢ Ⅎ𝑥 𝑦 = ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
6		ineq2 4114	. . . . 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 3018	. . . 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 3279	. 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 2763	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}
11		eqid 2763	. . 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 9964	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 {𝑧 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝐵 (rank‘𝑧) ⊆ (rank‘𝑤)}) ≠ ∅)
13	3, 9, 12	ceqsexv2d 3503	1 ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2959  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  ∪ ciun 4954  ‘cfv 6547  rankcrnk 9780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-reg 9599  ax-inf2 9655

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-r1 9781  df-rank 9782

This theorem is used by: (None)