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Theorem infenaleph 9506
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 488 . . . . . . . 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 486 . . . . . . . . 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 590 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
103, 9mpbird 260 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
112, 10eqsstrrid 3967 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
12 cardalephex 9505 . . . . . 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 236 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥))
15 eqcom 2808 . . . . 5 ((card‘𝐴) = (ℵ‘𝑥) ↔ (ℵ‘𝑥) = (card‘𝐴))
1615rexbii 3213 . . . 4 (∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥) ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
1714, 16sylib 221 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
18 alephfnon 9480 . . . 4 ℵ Fn On
19 fvelrnb 6705 . . . 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 237 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ∈ ran ℵ)
22 cardid2 9370 . . 3 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2322adantr 484 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
24 breq1 5036 . . 3 (𝑥 = (card‘𝐴) → (𝑥𝐴 ↔ (card‘𝐴) ≈ 𝐴))
2524rspcev 3574 . 2 (((card‘𝐴) ∈ ran ℵ ∧ (card‘𝐴) ≈ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥𝐴)
2621, 23, 25syl2anc 587 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wrex 3110  wss 3884   class class class wbr 5033  dom cdm 5523  ran crn 5524  Oncon0 6163   Fn wfn 6323  cfv 6328  ωcom 7564  cen 8493  cdom 8494  cardccrd 9352  cale 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-har 9009  df-card 9356  df-aleph 9357
This theorem is referenced by: (None)
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