Theorem cflem 9711

Description: A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set 𝐴. (Contributed by NM, 24-Apr-2004.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝐴

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

Proof of Theorem cflem

Step	Hyp	Ref	Expression
1		ssid 3916	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		ssid 3916	. . . . 5 ⊢ 𝑧 ⊆ 𝑧
3		sseq2 3920	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧))
4	3	rspcev 3543	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
5	2, 4	mpan2 690	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
6	5	rgen 3080	. . 3 ⊢ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤
7		sseq1 3919	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
8		rexeq 3324	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
9	8	ralbidv 3126	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
10	7, 9	anbi12d 633	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
11	10	spcegv 3517	. . 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 697	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
13		fvex 6675	. . . . . 6 ⊢ (card‘𝑦) ∈ V
14	13	isseti 3424	. . . . 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 708	. . . 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 2166	. . 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 280	. 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 221	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ⊆ wss 3860  ‘cfv 6339  cardccrd 9402

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 2729  ax-nul 5179

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-uni 4802  df-iota 6298  df-fv 6347

This theorem is referenced by:  cfval  9712  cff  9713  cff1  9723