Theorem cflemOLD 10137

Description: Obsolete version of cflem 10136 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 3952	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		ssid 3952	. . . . 5 ⊢ 𝑧 ⊆ 𝑧
3		sseq2 3956	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧))
4	3	rspcev 3572	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
5	2, 4	mpan2 691	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
6	5	rgen 3049	. . 3 ⊢ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤
7		sseq1 3955	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
8		rexeq 3288	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
9	8	ralbidv 3155	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
10	7, 9	anbi12d 632	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
11	10	spcegv 3547	. . 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 698	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
13		fvex 6835	. . . . . 6 ⊢ (card‘𝑦) ∈ V
14	13	isseti 3454	. . . . 5 ⊢ ∃𝑥 𝑥 = (card‘𝑦)
15		19.41v 1950	. . . . 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 709	. . . 4 ⊢ (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
17	16	exbii 1849	. . 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 2165	. . 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 277	. 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 218	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  ‘cfv 6481  cardccrd 9828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-sn 4574  df-pr 4576  df-uni 4857  df-iota 6437  df-fv 6489

This theorem is referenced by: (None)