Theorem cardlim 3657

Description: An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91.

Assertion

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

Proof of Theorem cardlim

Step	Hyp	Ref	Expression
1		sseq2 1522	. . . . . . . . . . 11 ⊢ ((card ‘A) = suc x → (ω ⊆ (card ‘A) ↔ ω ⊆ suc x))
2	1	biimpd 135	. . . . . . . . . 10 ⊢ ((card ‘A) = suc x → (ω ⊆ (card ‘A) → ω ⊆ suc x))
3		infensuc 3484	. . . . . . . . . . . 12 ⊢ ((x ∈ On ∧ ω ⊆ x) → x ≈ suc x)
4	3	exp 291	. . . . . . . . . . 11 ⊢ (x ∈ On → (ω ⊆ x → x ≈ suc x))
5		limom 2387	. . . . . . . . . . . 12 ⊢ Lim ω
6		limsssuc 2362	. . . . . . . . . . . 12 ⊢ (Lim ω → (ω ⊆ x ↔ ω ⊆ suc x))
7	5, 6	ax-mp 6	. . . . . . . . . . 11 ⊢ (ω ⊆ x ↔ ω ⊆ suc x)
8	4, 7	syl5ibr 182	. . . . . . . . . 10 ⊢ (x ∈ On → (ω ⊆ suc x → x ≈ suc x))
9	2, 8	sylan9r 360	. . . . . . . . 9 ⊢ ((x ∈ On ∧ (card ‘A) = suc x) → (ω ⊆ (card ‘A) → x ≈ suc x))
10		breq2 2066	. . . . . . . . . 10 ⊢ ((card ‘A) = suc x → (x ≈ (card ‘A) ↔ x ≈ suc x))
11	10	adantl 305	. . . . . . . . 9 ⊢ ((x ∈ On ∧ (card ‘A) = suc x) → (x ≈ (card ‘A) ↔ x ≈ suc x))
12	9, 11	sylibrd 179	. . . . . . . 8 ⊢ ((x ∈ On ∧ (card ‘A) = suc x) → (ω ⊆ (card ‘A) → x ≈ (card ‘A)))
13	12	exp 291	. . . . . . 7 ⊢ (x ∈ On → ((card ‘A) = suc x → (ω ⊆ (card ‘A) → x ≈ (card ‘A))))
14	13	com3r 35	. . . . . 6 ⊢ (ω ⊆ (card ‘A) → (x ∈ On → ((card ‘A) = suc x → x ≈ (card ‘A))))
15	14	imp 277	. . . . 5 ⊢ ((ω ⊆ (card ‘A) ∧ x ∈ On) → ((card ‘A) = suc x → x ≈ (card ‘A)))
16		visset 1350	. . . . . . . . . 10 ⊢ x ∈ V
17	16	sucid 2304	. . . . . . . . 9 ⊢ x ∈ suc x
18		eleq2 1150	. . . . . . . . 9 ⊢ ((card ‘A) = suc x → (x ∈ (card ‘A) ↔ x ∈ suc x))
19	17, 18	mpbiri 169	. . . . . . . 8 ⊢ ((card ‘A) = suc x → x ∈ (card ‘A))
20		cardcard 3655	. . . . . . . . 9 ⊢ (card ‘(card ‘A)) = (card ‘A)
21	20	eleq2i 1153	. . . . . . . 8 ⊢ (x ∈ (card ‘(card ‘A)) ↔ x ∈ (card ‘A))
22	19, 21	sylibr 175	. . . . . . 7 ⊢ ((card ‘A) = suc x → x ∈ (card ‘(card ‘A)))
23		cardne 3637	. . . . . . 7 ⊢ (x ∈ (card ‘(card ‘A)) → ¬ x ≈ (card ‘A))
24	22, 23	syl 12	. . . . . 6 ⊢ ((card ‘A) = suc x → ¬ x ≈ (card ‘A))
25	24	a1i 7	. . . . 5 ⊢ ((ω ⊆ (card ‘A) ∧ x ∈ On) → ((card ‘A) = suc x → ¬ x ≈ (card ‘A)))
26	15, 25	pm2.65d 117	. . . 4 ⊢ ((ω ⊆ (card ‘A) ∧ x ∈ On) → ¬ (card ‘A) = suc x)
27	26	nrexdv 1271	. . 3 ⊢ (ω ⊆ (card ‘A) → ¬ ∃x ∈ On (card ‘A) = suc x)
28		peano1 2390	. . . . . 6 ⊢ ∅ ∈ ω
29		ssel 1502	. . . . . 6 ⊢ (ω ⊆ (card ‘A) → (∅ ∈ ω → ∅ ∈ (card ‘A)))
30	28, 29	mpi 44	. . . . 5 ⊢ (ω ⊆ (card ‘A) → ∅ ∈ (card ‘A))
31		n0i 1712	. . . . 5 ⊢ (∅ ∈ (card ‘A) → ¬ (card ‘A) = ∅)
32		cardon 3634	. . . . . . . . 9 ⊢ (card ‘A) ∈ On
33	32	onord 2343	. . . . . . . 8 ⊢ Ord (card ‘A)
34		ordzsl 2366	. . . . . . . 8 ⊢ (Ord (card ‘A) ↔ ((card ‘A) = ∅ ∨ ∃x ∈ On (card ‘A) = suc x ∨ Lim (card ‘A)))
35	33, 34	mpbi 164	. . . . . . 7 ⊢ ((card ‘A) = ∅ ∨ ∃x ∈ On (card ‘A) = suc x ∨ Lim (card ‘A))
36		3orass 584	. . . . . . 7 ⊢ (((card ‘A) = ∅ ∨ ∃x ∈ On (card ‘A) = suc x ∨ Lim (card ‘A)) ↔ ((card ‘A) = ∅ ∨ (∃x ∈ On (card ‘A) = suc x ∨ Lim (card ‘A))))
37	35, 36	mpbi 164	. . . . . 6 ⊢ ((card ‘A) = ∅ ∨ (∃x ∈ On (card ‘A) = suc x ∨ Lim (card ‘A)))
38	37	ori 200	. . . . 5 ⊢ (¬ (card ‘A) = ∅ → (∃x ∈ On (card ‘A) = suc x ∨ Lim (card ‘A)))
39	30, 31, 38	3syl 21	. . . 4 ⊢ (ω ⊆ (card ‘A) → (∃x ∈ On (card ‘A) = suc x ∨ Lim (card ‘A)))
40	39	ord 202	. . 3 ⊢ (ω ⊆ (card ‘A) → (¬ ∃x ∈ On (card ‘A) = suc x → Lim (card ‘A)))
41	27, 40	mpd 46	. 2 ⊢ (ω ⊆ (card ‘A) → Lim (card ‘A))
42		limomss 2378	. 2 ⊢ (Lim (card ‘A) → ω ⊆ (card ‘A))
43	41, 42	impbi 139	1 ⊢ (ω ⊆ (card ‘A) ↔ Lim (card ‘A))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∨ w3o 580   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201  ωcom 2372   ‘cfv 2422   ≈ cen 3271  cardccrd 3620

This theorem is referenced by:  alephislim 3688

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-1o 3104  df-er 3200  df-en 3274  df-dom 3275  df-card 3623

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