Theorem cardlim 9865

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 3956	. . . . . . . . . . 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 7812	. . . . . . . . . . . 12 ⊢ Lim ω
4		limsssuc 7780	. . . . . . . . . . . 12 ⊢ (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6		infensuc 9068	. . . . . . . . . . . 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 5093	. . . . . . . . . 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 3440	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucid 6390	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
18		eleq2 2820	. . . . . . . . 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 9852	. . . . . . . 8 ⊢ (card‘(card‘𝐴)) = (card‘𝐴)
21	19, 20	eleqtrrdi 2842	. . . . . . 7 ⊢ ((card‘𝐴) = suc 𝑥 → 𝑥 ∈ (card‘(card‘𝐴)))
22		cardne 9858	. . . . . . 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 3127	. . 3 ⊢ (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27		peano1 7819	. . . . . 6 ⊢ ∅ ∈ ω
28		ssel 3923	. . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
29	27, 28	mpi 20	. . . . 5 ⊢ (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30		n0i 4287	. . . . 5 ⊢ (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31		cardon 9837	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
32	31	onordi 6419	. . . . . . . 8 ⊢ Ord (card‘𝐴)
33		ordzsl 7775	. . . . . . . 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 1089	. . . . . . 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 861	. . . . 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 864	. . 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 7801	. 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 847   ∨ w3o 1085   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897  ∅c0 4280   class class class wbr 5089  Ord word 6305  Oncon0 6306  Lim wlim 6307  suc csuc 6308  ‘cfv 6481  ωcom 7796   ≈ cen 8866  cardccrd 9828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-er 8622  df-en 8870  df-dom 8871  df-card 9832

This theorem is referenced by:  infxpenlem  9904  alephislim  9974  cflim2  10154  winalim  10586  gruina  10709