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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  ℵcale 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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