Theorem cardlim 9967

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 4009	. . . . . . . . . . 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 7871	. . . . . . . . . . . 12 ⊢ Lim ω
4		limsssuc 7839	. . . . . . . . . . . 12 ⊢ (Lim ω → (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ (ω ⊆ 𝑥 ↔ ω ⊆ suc 𝑥)
6		infensuc 9155	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
7	6	ex 414	. . . . . . . . . . 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 510	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (card‘𝐴) = suc 𝑥) → (ω ⊆ (card‘𝐴) → 𝑥 ≈ suc 𝑥))
10		breq2 5153	. . . . . . . . . 10 ⊢ ((card‘𝐴) = suc 𝑥 → (𝑥 ≈ (card‘𝐴) ↔ 𝑥 ≈ suc 𝑥))
11	10	adantl 483	. . . . . . . . 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 414	. . . . . . 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 408	. . . . 5 ⊢ ((ω ⊆ (card‘𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = suc 𝑥 → 𝑥 ≈ (card‘𝐴)))
16		vex 3479	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucid 6447	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
18		eleq2 2823	. . . . . . . . 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 9954	. . . . . . . 8 ⊢ (card‘(card‘𝐴)) = (card‘𝐴)
21	19, 20	eleqtrrdi 2845	. . . . . . 7 ⊢ ((card‘𝐴) = suc 𝑥 → 𝑥 ∈ (card‘(card‘𝐴)))
22		cardne 9960	. . . . . . 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 3150	. . 3 ⊢ (ω ⊆ (card‘𝐴) → ¬ ∃𝑥 ∈ On (card‘𝐴) = suc 𝑥)
27		peano1 7879	. . . . . 6 ⊢ ∅ ∈ ω
28		ssel 3976	. . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → (∅ ∈ ω → ∅ ∈ (card‘𝐴)))
29	27, 28	mpi 20	. . . . 5 ⊢ (ω ⊆ (card‘𝐴) → ∅ ∈ (card‘𝐴))
30		n0i 4334	. . . . 5 ⊢ (∅ ∈ (card‘𝐴) → ¬ (card‘𝐴) = ∅)
31		cardon 9939	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
32	31	onordi 6476	. . . . . . . 8 ⊢ Ord (card‘𝐴)
33		ordzsl 7834	. . . . . . . 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 1091	. . . . . . 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 860	. . . . 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 863	. . 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 7860	. 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 397   ∨ wo 846   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107  ∃wrex 3071   ⊆ wss 3949  ∅c0 4323   class class class wbr 5149  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  ‘cfv 6544  ωcom 7855   ≈ cen 8936  cardccrd 9930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-er 8703  df-en 8940  df-dom 8941  df-card 9934

This theorem is referenced by:  infxpenlem  10008  alephislim  10078  cflim2  10258  winalim  10690  gruina  10813