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Theorem infenaleph 8911
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 8782 . . . . 5 (card‘(card‘𝐴)) = (card‘𝐴)
2 cardom 8809 . . . . . . 7 (card‘ω) = ω
3 simpr 477 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴)
4 omelon 8540 . . . . . . . . . 10 ω ∈ On
5 onenon 8772 . . . . . . . . . 10 (ω ∈ On → ω ∈ dom card)
64, 5ax-mp 5 . . . . . . . . 9 ω ∈ dom card
7 simpl 473 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
8 carddom2 8800 . . . . . . . . 9 ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
96, 7, 8sylancr 695 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴))
103, 9mpbird 247 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴))
112, 10syl5eqssr 3648 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴))
12 cardalephex 8910 . . . . . 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 223 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥))
15 eqcom 2628 . . . . 5 ((card‘𝐴) = (ℵ‘𝑥) ↔ (ℵ‘𝑥) = (card‘𝐴))
1615rexbii 3039 . . . 4 (∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥) ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
1714, 16sylib 208 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))
18 alephfnon 8885 . . . 4 ℵ Fn On
19 fvelrnb 6241 . . . 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 224 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ∈ ran ℵ)
22 cardid2 8776 . . 3 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2322adantr 481 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴)
24 breq1 4654 . . 3 (𝑥 = (card‘𝐴) → (𝑥𝐴 ↔ (card‘𝐴) ≈ 𝐴))
2524rspcev 3307 . 2 (((card‘𝐴) ∈ ran ℵ ∧ (card‘𝐴) ≈ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥𝐴)
2621, 23, 25syl2anc 693 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wrex 2912  wss 3572   class class class wbr 4651  dom cdm 5112  ran crn 5113  Oncon0 5721   Fn wfn 5881  cfv 5886  ωcom 7062  cen 7949  cdom 7950  cardccrd 8758  cale 8759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-oi 8412  df-har 8460  df-card 8762  df-aleph 8763
This theorem is referenced by: (None)
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