Theorem cardlim 9925

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 3973	. . . . . . . . . . 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 7858	. . . . . . . . . . . 12 ⊢ Lim ω
4		limsssuc 7826	. . . . . . . . . . . 12 ⊢ (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6		infensuc 9119	. . . . . . . . . . . 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 5111	. . . . . . . . . 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 3451	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucid 6416	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
18		eleq2 2817	. . . . . . . . 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 9912	. . . . . . . 8 ⊢ (card‘(card‘𝐴)) = (card‘𝐴)
21	19, 20	eleqtrrdi 2839	. . . . . . 7 ⊢ ((card‘𝐴) = suc 𝑥 → 𝑥 ∈ (card‘(card‘𝐴)))
22		cardne 9918	. . . . . . 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 3128	. . 3 ⊢ (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27		peano1 7865	. . . . . 6 ⊢ ∅ ∈ ω
28		ssel 3940	. . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
29	27, 28	mpi 20	. . . . 5 ⊢ (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30		n0i 4303	. . . . 5 ⊢ (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31		cardon 9897	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
32	31	onordi 6445	. . . . . . . 8 ⊢ Ord (card‘𝐴)
33		ordzsl 7821	. . . . . . . 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 7847	. 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 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3914  ∅c0 4296   class class class wbr 5107  Ord word 6331  Oncon0 6332  Lim wlim 6333  suc csuc 6334  ‘cfv 6511  ωcom 7842   ≈ cen 8915  cardccrd 9888

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-er 8671  df-en 8919  df-dom 8920  df-card 9892

This theorem is referenced by:  infxpenlem  9966  alephislim  10036  cflim2  10216  winalim  10648  gruina  10771