Theorem alephinit 9851

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 9848	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
2	1	bicomi 223	. . . 4 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
3	2	baib 536	. . 3 ⊢ (ω ⊆ 𝐴 → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
4	3	adantl 482	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
5		onenon 9707	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
6	5	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7		onenon 9707	. . . . . . 7 ⊢ (𝑥 ∈ On → 𝑥 ∈ dom card)
8		carddom2 9735	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴 ≼ 𝑥))
9	6, 7, 8	syl2an 596	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴 ≼ 𝑥))
10		cardonle 9715	. . . . . . . 8 ⊢ (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
11	10	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12		sstr 3929	. . . . . . . 8 ⊢ (((card‘𝐴) ⊆ (card‘𝑥) ∧ (card‘𝑥) ⊆ 𝑥) → (card‘𝐴) ⊆ 𝑥)
13	12	expcom 414	. . . . . . 7 ⊢ ((card‘𝑥) ⊆ 𝑥 → ((card‘𝐴) ⊆ (card‘𝑥) → (card‘𝐴) ⊆ 𝑥))
14	11, 13	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) → (card‘𝐴) ⊆ 𝑥))
15	9, 14	sylbird 259	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (𝐴 ≼ 𝑥 → (card‘𝐴) ⊆ 𝑥))
16		sseq1 3946	. . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
17	16	imbi2d 341	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → ((𝐴 ≼ 𝑥 → (card‘𝐴) ⊆ 𝑥) ↔ (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
18	15, 17	syl5ibcom 244	. . . 4 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = 𝐴 → (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
19	18	ralrimdva 3106	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
20		oncardid 9714	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21		ensym 8789	. . . . . . 7 ⊢ ((card‘𝐴) ≈ 𝐴 → 𝐴 ≈ (card‘𝐴))
22		endom 8767	. . . . . . 7 ⊢ (𝐴 ≈ (card‘𝐴) → 𝐴 ≼ (card‘𝐴))
23	20, 21, 22	3syl 18	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ≼ (card‘𝐴))
24	23	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≼ (card‘𝐴))
25		cardon 9702	. . . . . 6 ⊢ (card‘𝐴) ∈ On
26		breq2 5078	. . . . . . . 8 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≼ 𝑥 ↔ 𝐴 ≼ (card‘𝐴)))
27		sseq2 3947	. . . . . . . 8 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (card‘𝐴)))
28	26, 27	imbi12d 345	. . . . . . 7 ⊢ (𝑥 = (card‘𝐴) → ((𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) ↔ (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
29	28	rspcv 3557	. . . . . 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 31	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) → 𝐴 ⊆ (card‘𝐴)))
32		cardonle 9715	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
33	32	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘𝐴) ⊆ 𝐴)
34	33	biantrurd 533	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))))
35		eqss 3936	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴)))
36	34, 35	bitr4di 289	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
37	31, 36	sylibd 238	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) → (card‘𝐴) = 𝐴))
38	19, 37	impbid 211	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
39	4, 38	bitrd 278	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887   class class class wbr 5074  dom cdm 5589  ran crn 5590  Oncon0 6266  ‘cfv 6433  ωcom 7712   ≈ cen 8730   ≼ cdom 8731  cardccrd 9693  ℵcale 9694

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-har 9316  df-card 9697  df-aleph 9698

This theorem is referenced by: (None)