Theorem infenaleph 10004

Description: An infinite numerable set is equinumerous to an infinite initial ordinal. (Contributed by Jeff Hankins, 23-Oct-2009.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infenaleph	⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem infenaleph

Step	Hyp	Ref	Expression
1		cardidm 9874	. . . . 5 ⊢ (card‘(card‘𝐴)) = (card‘𝐴)
2		cardom 9901	. . . . . . 7 ⊢ (card‘ω) = ω
3		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
4		omelon 9561	. . . . . . . . . 10 ⊢ ω ∈ On
5		onenon 9864	. . . . . . . . . 10 ⊢ (ω ∈ On → ω ∈ dom card)
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ω ∈ dom card
7		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
8		carddom2 9892	. . . . . . . . 9 ⊢ ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
9	6, 7, 8	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
10	3, 9	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
11	2, 10	eqsstrrid 3977	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
12		cardalephex 10003	. . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → ((card‘(card‘𝐴)) = (card‘𝐴) ↔ ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥)))
13	11, 12	syl 17	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘(card‘𝐴)) = (card‘𝐴) ↔ ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥)))
14	1, 13	mpbii 233	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥))
15		eqcom 2736	. . . . 5 ⊢ ((card‘𝐴) = (ℵ‘𝑥) ↔ (ℵ‘𝑥) = (card‘𝐴))
16	15	rexbii 3076	. . . 4 ⊢ (∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥) ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
17	14, 16	sylib 218	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
18		alephfnon 9978	. . . 4 ⊢ ℵ Fn On
19		fvelrnb 6887	. . . 4 ⊢ (ℵ Fn On → ((card‘𝐴) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴)))
20	18, 19	ax-mp 5	. . 3 ⊢ ((card‘𝐴) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
21	17, 20	sylibr 234	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ∈ ran ℵ)
22		cardid2 9868	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
23	22	adantr 480	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
24		breq1 5098	. . 3 ⊢ (𝑥 = (card‘𝐴) → (𝑥 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴))
25	24	rspcev 3579	. 2 ⊢ (((card‘𝐴) ∈ ran ℵ ∧ (card‘𝐴) ≈ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴)
26	21, 23, 25	syl2anc 584	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3905   class class class wbr 5095  dom cdm 5623  ran crn 5624  Oncon0 6311   Fn wfn 6481  ‘cfv 6486  ωcom 7806   ≈ cen 8876   ≼ cdom 8877  cardccrd 9850  ℵcale 9851

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556

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 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-oi 9421  df-har 9468  df-card 9854  df-aleph 9855

This theorem is referenced by: (None)