Theorem cardlim 10012

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 4010	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ suc 𝑥))
2	1	biimpd 229	. . . . . . . . . 10 ⊢ ((card‘𝐴) = suc 𝑥 → (ω ⊆ (card‘𝐴) → ω ⊆ suc 𝑥))
3		limom 7903	. . . . . . . . . . . 12 ⊢ Lim ω
4		limsssuc 7871	. . . . . . . . . . . 12 ⊢ (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6		infensuc 9195	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
7	6	ex 412	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (ω ⊆ 𝑥 → 𝑥 ≈ suc 𝑥))
8	5, 7	biimtrrid 243	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (ω ⊆ suc 𝑥 → 𝑥 ≈ suc 𝑥))
9	2, 8	sylan9r 508	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ suc 𝑥))
10		breq2 5147	. . . . . . . . . 10 ⊢ ((card‘𝐴) = suc 𝑥 → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
11	10	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
12	9, 11	sylibrd 259	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ (card‘𝐴)))
13	12	ex 412	. . . . . . 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 406	. . . . 5 ⊢ ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥 → 𝑥 ≈ (card‘𝐴)))
16		vex 3484	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucid 6466	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
18		eleq2 2830	. . . . . . . . 9 ⊢ ((card‘𝐴) = suc 𝑥 → (𝑥 ∈ (card‘𝐴) ↔ 𝑥 ∈ suc 𝑥))
19	17, 18	mpbiri 258	. . . . . . . 8 ⊢ ((card‘𝐴) = suc 𝑥 → 𝑥 ∈ (card‘𝐴))
20		cardidm 9999	. . . . . . . 8 ⊢ (card‘(card‘𝐴)) = (card‘𝐴)
21	19, 20	eleqtrrdi 2852	. . . . . . 7 ⊢ ((card‘𝐴) = suc 𝑥 → 𝑥 ∈ (card‘(card‘𝐴)))
22		cardne 10005	. . . . . . 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 196	. . . 4 ⊢ ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ¬ (card‘𝐴) = suc 𝑥)
26	25	nrexdv 3149	. . 3 ⊢ (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27		peano1 7910	. . . . . 6 ⊢ ∅ ∈ ω
28		ssel 3977	. . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
29	27, 28	mpi 20	. . . . 5 ⊢ (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30		n0i 4340	. . . . 5 ⊢ (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31		cardon 9984	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
32	31	onordi 6495	. . . . . . . 8 ⊢ Ord (card‘𝐴)
33		ordzsl 7866	. . . . . . . 8 ⊢ (Ord (card‘𝐴) ↔ ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
34	32, 33	mpbi 230	. . . . . . 7 ⊢ ((card‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴))
35		3orass 1090	. . . . . . 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 230	. . . . . 6 ⊢ ((card‘𝐴) = ∅ ∨ (∃𝑥 ∈ On (card‘𝐴) = suc 𝑥 ∨ Lim (card‘𝐴)))
37	36	ori 862	. . . . 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 865	. . 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 7892	. 2 ⊢ (Lim (card‘𝐴) → ω ⊆ (card‘𝐴))
42	40, 41	impbii 209	1 ⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  ‘cfv 6561  ωcom 7887   ≈ cen 8982  cardccrd 9975

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-er 8745  df-en 8986  df-dom 8987  df-card 9979

This theorem is referenced by:  infxpenlem  10053  alephislim  10123  cflim2  10303  winalim  10735  gruina  10858