Theorem infenaleph 10141

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 10011	. . . . 5 ⊢ (card‘(card‘𝐴)) = (card‘𝐴)
2		cardom 10038	. . . . . . 7 ⊢ (card‘ω) = ω
3		simpr 490	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
4		omelon 9625	. . . . . . . . . 10 ⊢ ω ∈ On
5		onenon 10001	. . . . . . . . . 10 ⊢ (ω ∈ On → ω ∈ dom card)
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ω ∈ dom card
7		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
8		carddom2 10029	. . . . . . . . 9 ⊢ ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
9	6, 7, 8	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
10	3, 9	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
11	2, 10	eqsstrrid 3969	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
12		cardalephex 10140	. . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → ((card‘(card‘𝐴)) = (card‘𝐴) ↔ ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥)))
13	11, 12	syl 18	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘(card‘𝐴)) = (card‘𝐴) ↔ ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥)))
14	1, 13	mpbii 236	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥))
15		eqcom 2767	. . . . 5 ⊢ ((card‘𝐴) = (ℵ‘𝑥) ↔ (ℵ‘𝑥) = (card‘𝐴))
16	15	rexbii 3109	. . . 4 ⊢ (∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥) ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
17	14, 16	sylib 221	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
18		alephfnon 10115	. . . 4 ⊢ ℵ Fn On
19		fvelrnb 6933	. . . 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 237	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ∈ ran ℵ)
22		cardid2 10005	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
23	22	adantr 486	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
24		breq1 5105	. . 3 ⊢ (𝑥 = (card‘𝐴) → (𝑥 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴))
25	24	rspcev 3576	. 2 ⊢ (((card‘𝐴) ∈ ran ℵ ∧ (card‘𝐴) ≈ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴)
26	21, 23, 25	syl2anc 596	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3898   class class class wbr 5102  dom cdm 5647  ran crn 5648  Oncon0 6351   Fn wfn 6522  ‘cfv 6527  ωcom 7860   ≈ cen 8948   ≼ cdom 8949  cardccrd 9987  ℵcale 9988

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-oi 9482  df-har 9529  df-card 9991  df-aleph 9992

This theorem is used by: (None)