Theorem alephinit 10101

Description: An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
alephinit	⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem alephinit

Step	Hyp	Ref	Expression
1		isinfcard 10098	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
2	1	bicomi 227	. . . 4 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
3	2	baib 545	. . 3 ⊢ (ω ⊆ 𝐴 → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
4	3	adantl 487	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
5		onenon 9957	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
6	5	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7		onenon 9957	. . . . . . 7 ⊢ (𝑥 ∈ On → 𝑥 ∈ dom card)
8		carddom2 9985	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴 ≼ 𝑥))
9	6, 7, 8	syl2an 608	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴 ≼ 𝑥))
10		cardonle 9965	. . . . . . . 8 ⊢ (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
11	10	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12		sstr 3942	. . . . . . . 8 ⊢ (((card‘𝐴) ⊆ (card‘𝑥) ∧ (card‘𝑥) ⊆ 𝑥) → (card‘𝐴) ⊆ 𝑥)
13	12	expcom 419	. . . . . . 7 ⊢ ((card‘𝑥) ⊆ 𝑥 → ((card‘𝐴) ⊆ (card‘𝑥) → (card‘𝐴) ⊆ 𝑥))
14	11, 13	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) → (card‘𝐴) ⊆ 𝑥))
15	9, 14	sylbird 263	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (𝐴 ≼ 𝑥 → (card‘𝐴) ⊆ 𝑥))
16		sseq1 3959	. . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
17	16	imbi2d 343	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → ((𝐴 ≼ 𝑥 → (card‘𝐴) ⊆ 𝑥) ↔ (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
18	15, 17	syl5ibcom 248	. . . 4 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = 𝐴 → (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
19	18	ralrimdva 3164	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
20		oncardid 9964	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21		ensym 9012	. . . . . . 7 ⊢ ((card‘𝐴) ≈ 𝐴 → 𝐴 ≈ (card‘𝐴))
22		endom 8988	. . . . . . 7 ⊢ (𝐴 ≈ (card‘𝐴) → 𝐴 ≼ (card‘𝐴))
23	20, 21, 22	3syl 19	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ≼ (card‘𝐴))
24	23	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≼ (card‘𝐴))
25		cardon 9952	. . . . . 6 ⊢ (card‘𝐴) ∈ On
26		breq2 5111	. . . . . . . 8 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≼ 𝑥 ↔ 𝐴 ≼ (card‘𝐴)))
27		sseq2 3960	. . . . . . . 8 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (card‘𝐴)))
28	26, 27	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = (card‘𝐴) → ((𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) ↔ (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
29	28	rspcv 3575	. . . . . 6 ⊢ ((card‘𝐴) ∈ On → (∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) → (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
30	25, 29	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) → (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴)))
31	24, 30	syl5com 32	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) → 𝐴 ⊆ (card‘𝐴)))
32		cardonle 9965	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
33	32	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘𝐴) ⊆ 𝐴)
34	33	biantrurd 542	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))))
35		eqss 3949	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴)))
36	34, 35	bitr4di 292	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
37	31, 36	sylibd 242	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) → (card‘𝐴) = 𝐴))
38	19, 37	impbid 215	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
39	4, 38	bitrd 282	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902   class class class wbr 5107  dom cdm 5659  ran crn 5660  Oncon0 6361  ‘cfv 6537  ωcom 7865   ≈ cen 8952   ≼ cdom 8953  cardccrd 9943  ℵcale 9944

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9485  df-har 9532  df-card 9947  df-aleph 9948

This theorem is used by: (None)