Theorem cardlim 9969

Description: An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)

Assertion

Ref	Expression
cardlim	⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))

Proof of Theorem cardlim

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 4007	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ suc 𝑥))
2	1	biimpd 228	. . . . . . . . . 10 ⊢ ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) → ω ⊆ suc 𝑥))
3		limom 7873	. . . . . . . . . . . 12 ⊢ Lim ω
4		limsssuc 7841	. . . . . . . . . . . 12 ⊢ (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6		infensuc 9157	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
7	6	ex 411	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (ω ⊆ 𝑥 → 𝑥 ≈ suc 𝑥))
8	5, 7	biimtrrid 242	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (ω ⊆ suc 𝑥 → 𝑥 ≈ suc 𝑥))
9	2, 8	sylan9r 507	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ suc 𝑥))
10		breq2 5151	. . . . . . . . . 10 ⊢ ((card‘𝐴) = suc 𝑥 → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
11	10	adantl 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
12	9, 11	sylibrd 258	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ (card‘𝐴)))
13	12	ex 411	. . . . . . 7 ⊢ (𝑥 ∈ On → ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) → 𝑥 ≈ (card‘𝐴))))
14	13	com3r 87	. . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → (𝑥 ∈ On → ((card‘𝐴) = suc 𝑥 → 𝑥 ≈ (card‘𝐴))))
15	14	imp 405	. . . . 5 ⊢ ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥 → 𝑥 ≈ (card‘𝐴)))
16		vex 3476	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucid 6445	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
18		eleq2 2820	. . . . . . . . 9 ⊢ ((card‘𝐴) = suc 𝑥 → (𝑥 ∈ (card‘𝐴) ↔ 𝑥 ∈ suc 𝑥))
19	17, 18	mpbiri 257	. . . . . . . 8 ⊢ ((card‘𝐴) = suc 𝑥 → 𝑥 ∈ (card‘𝐴))
20		cardidm 9956	. . . . . . . 8 ⊢ (card‘(card‘𝐴)) = (card‘𝐴)
21	19, 20	eleqtrrdi 2842	. . . . . . 7 ⊢ ((card‘𝐴) = suc 𝑥 → 𝑥 ∈ (card‘(card‘𝐴)))
22		cardne 9962	. . . . . . 7 ⊢ (𝑥 ∈ (card‘(card‘𝐴)) → ¬ 𝑥 ≈ (card‘𝐴))
23	21, 22	syl 17	. . . . . 6 ⊢ ((card‘𝐴) = suc 𝑥 → ¬ 𝑥 ≈ (card‘𝐴))
24	23	a1i 11	. . . . 5 ⊢ ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥 → ¬ 𝑥 ≈ (card‘𝐴)))
25	15, 24	pm2.65d 195	. . . 4 ⊢ ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ¬ (card‘𝐴) = suc 𝑥)
26	25	nrexdv 3147	. . 3 ⊢ (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27		peano1 7881	. . . . . 6 ⊢ ∅ ∈ ω
28		ssel 3974	. . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
29	27, 28	mpi 20	. . . . 5 ⊢ (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30		n0i 4332	. . . . 5 ⊢ (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31		cardon 9941	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
32	31	onordi 6474	. . . . . . . 8 ⊢ Ord (card‘𝐴)
33		ordzsl 7836	. . . . . . . 8 ⊢ (Ord (card‘𝐴) ↔ ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
34	32, 33	mpbi 229	. . . . . . 7 ⊢ ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴))
35		3orass 1088	. . . . . . 7 ⊢ (((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)) ↔ ((card‘𝐴) = ∅ ∨ (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴))))
36	34, 35	mpbi 229	. . . . . 6 ⊢ ((card‘𝐴) = ∅ ∨ (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
37	36	ori 857	. . . . 5 ⊢ (¬ (card‘𝐴) = ∅ → (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
38	29, 30, 37	3syl 18	. . . 4 ⊢ (ω ⊆ (card‘𝐴) → (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
39	38	ord 860	. . 3 ⊢ (ω ⊆ (card‘𝐴) → (¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 → Lim (card‘𝐴)))
40	26, 39	mpd 15	. 2 ⊢ (ω ⊆ (card‘𝐴) → Lim (card‘𝐴))
41		limomss 7862	. 2 ⊢ (Lim (card‘𝐴) → ω ⊆ (card‘𝐴))
42	40, 41	impbii 208	1 ⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∨ w3o 1084   = wceq 1539   ∈ wcel 2104  ∃wrex 3068   ⊆ wss 3947  ∅c0 4321   class class class wbr 5147  Ord word 6362  Oncon0 6363  Lim wlim 6364  suc csuc 6365  ‘cfv 6542  ωcom 7857   ≈ cen 8938  cardccrd 9932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-er 8705  df-en 8942  df-dom 8943  df-card 9936

This theorem is referenced by:  infxpenlem  10010  alephislim  10080  cflim2  10260  winalim  10692  gruina  10815