Theorem cflem 10326

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.) Avoid ax-11 2193. (Revised by BTernaryTau, 25-Jul-2025.)

Assertion

Ref	Expression
cflem	⊢ (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 cflem

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3930	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		ssid 3930	. . . . 5 ⊢ 𝑧 ⊆ 𝑧
3		sseq2 3934	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧))
4	3	rspcev 3576	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
5	2, 4	mpan2 704	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
6	5	rgen 3078	. . 3 ⊢ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤
7		sseq1 3933	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
8		rexeq 3315	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
9	8	ralbidv 3185	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
10	7, 9	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
11	10	spcegv 3551	. . 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 711	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
13		fvex 6898	. . . . . 6 ⊢ (card‘𝑦) ∈ V
14	13	isseti 3468	. . . . 5 ⊢ ∃𝑥 𝑥 = (card‘𝑦)
15		19.41v 1981	. . . . 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 722	. . . 4 ⊢ (∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
17	16	exbii 1880	. . 3 ⊢ (∃𝑦∃𝑥(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
18		fveq2 6885	. . . . . 6 ⊢ (𝑦 = 𝑣 → (card‘𝑦) = (card‘𝑣))
19	18	eqeq2d 2771	. . . . 5 ⊢ (𝑦 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝑣)))
20		sseq1 3933	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑦 ⊆ 𝐴 ↔ 𝑣 ⊆ 𝐴))
21		rexeq 3315	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑣 𝑧 ⊆ 𝑤))
22	21	ralbidv 3185	. . . . . 6 ⊢ (𝑦 = 𝑣 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑣 𝑧 ⊆ 𝑤))
23	20, 22	anbi12d 644	. . . . 5 ⊢ (𝑦 = 𝑣 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝑣 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑣 𝑧 ⊆ 𝑤)))
24	19, 23	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑥 = (card‘𝑣) ∧ (𝑣 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑣 𝑧 ⊆ 𝑤))))
25	24	excomimw 2076	. . 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)))
26	17, 25	sylbir 238	. 2 ⊢ (∃𝑦(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
27	12, 26	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ‘cfv 6538  cardccrd 10019

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546

This theorem is used by:  cfval  10327  cff  10328  cff1  10339