Theorem cflemOLD 10204

Description: Obsolete version of cflem 10203 as of 25-Jul-2025. (Contributed by NM, 24-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cflemOLD	⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))

Distinct variable group:   𝑤,𝐴,𝑥,𝑦,𝑧

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cflemOLD

Step	Hyp	Ref	Expression
1		ssid 3960	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		ssid 3960	. . . . 5 ⊢ 𝑧 ⊆ 𝑧
3		sseq2 3964	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧))
4	3	rspcev 3583	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
5	2, 4	mpan2 701	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
6	5	rgen 3080	. . 3 ⊢ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤
7		sseq1 3963	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
8		rexeq 3318	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
9	8	ralbidv 3187	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
10	7, 9	anbi12d 641	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
11	10	spcegv 3558	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
12	1, 6, 11	mp2ani 708	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
13		fvex 6882	. . . . . 6 ⊢ (card‘𝑦) ∈ V
14	13	isseti 3474	. . . . 5 ⊢ ∃𝑥 𝑥 = (card‘𝑦)
15		19.41v 1971	. . . . 5 ⊢ (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (∃𝑥 𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
16	14, 15	mpbiran 719	. . . 4 ⊢ (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
17	16	exbii 1870	. . 3 ⊢ (∃𝑦∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
18		excom 2198	. . 3 ⊢ (∃𝑦∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
19	17, 18	bitr3i 279	. 2 ⊢ (∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
20	12, 19	sylib 220	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088   ⊆ wss 3906  ‘cfv 6523  cardccrd 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6479  df-fv 6531

This theorem is referenced by: (None)