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Theorem infenaleph 9505
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 ℵ𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem infenaleph
StepHypRef Expression
1 cardidm 9376 . . . . 5 (card‘(card‘𝐴)) = (card‘𝐴)
2 cardom 9403 . . . . . . 7 (card‘ω) = ω
3 simpr 485 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
4 omelon 9097 . . . . . . . . . 10 ω ∈ On
5 onenon 9366 . . . . . . . . . 10 (ω ∈ On → ω ∈ dom card)
64, 5ax-mp 5 . . . . . . . . 9 ω ∈ dom card
7 simpl 483 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
8 carddom2 9394 . . . . . . . . 9 ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
96, 7, 8sylancr 587 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
103, 9mpbird 258 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
112, 10eqsstrrid 4013 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
12 cardalephex 9504 . . . . . 6 (ω ⊆ (card‘𝐴) → ((card‘(card‘𝐴)) = (card‘𝐴) ↔ ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥)))
1311, 12syl 17 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘(card‘𝐴)) = (card‘𝐴) ↔ ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥)))
141, 13mpbii 234 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥))
15 eqcom 2825 . . . . 5 ((card‘𝐴) = (ℵ‘𝑥) ↔ (ℵ‘𝑥) = (card‘𝐴))
1615rexbii 3244 . . . 4 (∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥) ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
1714, 16sylib 219 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
18 alephfnon 9479 . . . 4 ℵ Fn On
19 fvelrnb 6719 . . . 4 (ℵ Fn On → ((card‘𝐴) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴)))
2018, 19ax-mp 5 . . 3 ((card‘𝐴) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
2117, 20sylibr 235 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ∈ ran ℵ)
22 cardid2 9370 . . 3 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2322adantr 481 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
24 breq1 5060 . . 3 (𝑥 = (card‘𝐴) → (𝑥𝐴 ↔ (card‘𝐴) ≈ 𝐴))
2524rspcev 3620 . 2 (((card‘𝐴) ∈ ran ℵ ∧ (card‘𝐴) ≈ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥𝐴)
2621, 23, 25syl2anc 584 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  wss 3933   class class class wbr 5057  dom cdm 5548  ran crn 5549  Oncon0 6184   Fn wfn 6343  cfv 6348  ωcom 7569  cen 8494  cdom 8495  cardccrd 9352  cale 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-har 9010  df-card 9356  df-aleph 9357
This theorem is referenced by: (None)
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