Theorem alephinit 10054

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 10051	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
2	1	bicomi 224	. . . 4 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
3	2	baib 535	. . 3 ⊢ (ω ⊆ 𝐴 → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
4	3	adantl 481	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
5		onenon 9908	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7		onenon 9908	. . . . . . 7 ⊢ (𝑥 ∈ On → 𝑥 ∈ dom card)
8		carddom2 9936	. . . . . . 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 9916	. . . . . . . 8 ⊢ (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
11	10	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12		sstr 3957	. . . . . . . 8 ⊢ (((card‘𝐴) ⊆ (card‘𝑥) ∧ (card‘𝑥) ⊆ 𝑥) → (card‘𝐴) ⊆ 𝑥)
13	12	expcom 413	. . . . . . 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 260	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (𝐴 ≼ 𝑥 → (card‘𝐴) ⊆ 𝑥))
16		sseq1 3974	. . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
17	16	imbi2d 340	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → ((𝐴 ≼ 𝑥 → (card‘𝐴) ⊆ 𝑥) ↔ (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
18	15, 17	syl5ibcom 245	. . . 4 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = 𝐴 → (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
19	18	ralrimdva 3134	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
20		oncardid 9915	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21		ensym 8976	. . . . . . 7 ⊢ ((card‘𝐴) ≈ 𝐴 → 𝐴 ≈ (card‘𝐴))
22		endom 8952	. . . . . . 7 ⊢ (𝐴 ≈ (card‘𝐴) → 𝐴 ≼ (card‘𝐴))
23	20, 21, 22	3syl 18	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ≼ (card‘𝐴))
24	23	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≼ (card‘𝐴))
25		cardon 9903	. . . . . 6 ⊢ (card‘𝐴) ∈ On
26		breq2 5113	. . . . . . . 8 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≼ 𝑥 ↔ 𝐴 ≼ (card‘𝐴)))
27		sseq2 3975	. . . . . . . 8 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (card‘𝐴)))
28	26, 27	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = (card‘𝐴) → ((𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) ↔ (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
29	28	rspcv 3587	. . . . . 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 9916	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
33	32	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘𝐴) ⊆ 𝐴)
34	33	biantrurd 532	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))))
35		eqss 3964	. . . . 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 239	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥) → (card‘𝐴) = 𝐴))
38	19, 37	impbid 212	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
39	4, 38	bitrd 279	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3916   class class class wbr 5109  dom cdm 5640  ran crn 5641  Oncon0 6334  ‘cfv 6513  ωcom 7844   ≈ cen 8917   ≼ cdom 8918  cardccrd 9894  ℵcale 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-inf2 9600

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-isom 6522  df-riota 7346  df-ov 7392  df-om 7845  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-oi 9469  df-har 9516  df-card 9898  df-aleph 9899

This theorem is referenced by: (None)