Theorem cflem 3700

Description: A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set A.

Assertion

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

Distinct variable group(s):   x,y,z,w,A

Proof of Theorem cflem

Step	Hyp	Ref	Expression
1		ssid 1519	. . 3 ⊢ A ⊆ A
2		ssid 1519	. . . . 5 ⊢ z ⊆ z
3		sseq2 1522	. . . . . 6 ⊢ (w = z → (z ⊆ w ↔ z ⊆ z))
4	3	rcla4ev 1403	. . . . 5 ⊢ ((z ∈ A ∧ z ⊆ z) → ∃w ∈ A z ⊆ w)
5	2, 4	mpan2 519	. . . 4 ⊢ (z ∈ A → ∃w ∈ A z ⊆ w)
6	5	rgen 1247	. . 3 ⊢ ∀z ∈ A ∃w ∈ A z ⊆ w
7		sseq1 1521	. . . . 5 ⊢ (y = A → (y ⊆ A ↔ A ⊆ A))
8		rexeq 1325	. . . . . 6 ⊢ (y = A → (∃w ∈ y z ⊆ w ↔ ∃w ∈ A z ⊆ w))
9	8	biraldv 1219	. . . . 5 ⊢ (y = A → (∀z ∈ A ∃w ∈ y z ⊆ w ↔ ∀z ∈ A ∃w ∈ A z ⊆ w))
10	7, 9	anbi12d 476	. . . 4 ⊢ (y = A → ((y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w) ↔ (A ⊆ A ∧ ∀z ∈ A ∃w ∈ A z ⊆ w)))
11	10	cla4egv 1397	. . 3 ⊢ (A ∈ B → ((A ⊆ A ∧ ∀z ∈ A ∃w ∈ A z ⊆ w) → ∃y(y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)))
12	1, 6, 11	mp2ani 523	. 2 ⊢ (A ∈ B → ∃y(y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))
13		19.41v 963	. . . . 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)))
14		fvex 2838	. . . . . 6 ⊢ (card ‘y) ∈ V
15	14	isseti 1352	. . . . 5 ⊢ ∃x x = (card ‘y)
16	13, 15	mpbiran 547	. . . 4 ⊢ (∃x(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)) ↔ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))
17	16	biex 733	. . 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 728	. . 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	bitr3 153	. 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 173	1 ⊢ (A ∈ B → ∃x∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487   ‘cfv 2422  cardccrd 3620

This theorem is referenced by:  cfval 3701  cffnon 3702

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920  df-fv 2438

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