Theorem alephinit 9515

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 9512	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
2	1	bicomi 226	. . . 4 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
3	2	baib 538	. . 3 ⊢ (ω ⊆ 𝐴 → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
4	3	adantl 484	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
5		onenon 9372	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
6	5	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7		onenon 9372	. . . . . . 7 ⊢ (𝑥 ∈ On → 𝑥 ∈ dom card)
8		carddom2 9400	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴 ≼ 𝑥))
9	6, 7, 8	syl2an 597	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴 ≼ 𝑥))
10		cardonle 9380	. . . . . . . 8 ⊢ (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
11	10	adantl 484	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12		sstr 3975	. . . . . . . 8 ⊢ (((card‘𝐴) ⊆ (card‘𝑥) ∧ (card‘𝑥) ⊆ 𝑥) → (card‘𝐴) ⊆ 𝑥)
13	12	expcom 416	. . . . . . 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 262	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (𝐴 ≼ 𝑥 → (card‘𝐴) ⊆ 𝑥))
16		sseq1 3992	. . . . . 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 247	. . . 4 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = 𝐴 → (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
19	18	ralrimdva 3189	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
20		oncardid 9379	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21		ensym 8552	. . . . . . 7 ⊢ ((card‘𝐴) ≈ 𝐴 → 𝐴 ≈ (card‘𝐴))
22		endom 8530	. . . . . . 7 ⊢ (𝐴 ≈ (card‘𝐴) → 𝐴 ≼ (card‘𝐴))
23	20, 21, 22	3syl 18	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ≼ (card‘𝐴))
24	23	adantr 483	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≼ (card‘𝐴))
25		cardon 9367	. . . . . 6 ⊢ (card‘𝐴) ∈ On
26		breq2 5063	. . . . . . . 8 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≼ 𝑥 ↔ 𝐴 ≼ (card‘𝐴)))
27		sseq2 3993	. . . . . . . 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 3618	. . . . . 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 9380	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
33	32	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘𝐴) ⊆ 𝐴)
34	33	biantrurd 535	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))))
35		eqss 3982	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴)))
36	34, 35	syl6bbr 291	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
37	31, 36	sylibd 241	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) → (card‘𝐴) = 𝐴))
38	19, 37	impbid 214	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
39	4, 38	bitrd 281	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ⊆ wss 3936   class class class wbr 5059  dom cdm 5550  ran crn 5551  Oncon0 6186  ‘cfv 6350  ωcom 7574   ≈ cen 8500   ≼ cdom 8501  cardccrd 9358  ℵcale 9359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-oi 8968  df-har 9016  df-card 9362  df-aleph 9363

This theorem is referenced by: (None)