Theorem alephinit 10024

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 10021	. . . . 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 9878	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7		onenon 9878	. . . . . . 7 ⊢ (𝑥 ∈ On → 𝑥 ∈ dom card)
8		carddom2 9906	. . . . . . 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 9886	. . . . . . . 8 ⊢ (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
11	10	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12		sstr 3952	. . . . . . . 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 3969	. . . . . 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 3133	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴 ≼ 𝑥 → 𝐴 ⊆ 𝑥)))
20		oncardid 9885	. . . . . . 7 ⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21		ensym 8951	. . . . . . 7 ⊢ ((card‘𝐴) ≈ 𝐴 → 𝐴 ≈ (card‘𝐴))
22		endom 8927	. . . . . . 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 9873	. . . . . 6 ⊢ (card‘𝐴) ∈ On
26		breq2 5106	. . . . . . . 8 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≼ 𝑥 ↔ 𝐴 ≼ (card‘𝐴)))
27		sseq2 3970	. . . . . . . 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 3581	. . . . . 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 9886	. . . . . . 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 3959	. . . . 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 3044   ⊆ wss 3911   class class class wbr 5102  dom cdm 5631  ran crn 5632  Oncon0 6320  ‘cfv 6499  ωcom 7822   ≈ cen 8892   ≼ cdom 8893  cardccrd 9864  ℵcale 9865

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-oi 9439  df-har 9486  df-card 9868  df-aleph 9869

This theorem is referenced by: (None)