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Theorem alephinit 8878
Description: An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
alephinit ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem alephinit
StepHypRef Expression
1 isinfcard 8875 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
21bicomi 214 . . . 4 (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
32baib 943 . . 3 (ω ⊆ 𝐴 → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
43adantl 482 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
5 onenon 8735 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ dom card)
65adantr 481 . . . . . . 7 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7 onenon 8735 . . . . . . 7 (𝑥 ∈ On → 𝑥 ∈ dom card)
8 carddom2 8763 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝑥 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴𝑥))
96, 7, 8syl2an 494 . . . . . 6 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴𝑥))
10 cardonle 8743 . . . . . . . 8 (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
1110adantl 482 . . . . . . 7 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12 sstr 3596 . . . . . . . 8 (((card‘𝐴) ⊆ (card‘𝑥) ∧ (card‘𝑥) ⊆ 𝑥) → (card‘𝐴) ⊆ 𝑥)
1312expcom 451 . . . . . . 7 ((card‘𝑥) ⊆ 𝑥 → ((card‘𝐴) ⊆ (card‘𝑥) → (card‘𝐴) ⊆ 𝑥))
1411, 13syl 17 . . . . . 6 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) → (card‘𝐴) ⊆ 𝑥))
159, 14sylbird 250 . . . . 5 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (𝐴𝑥 → (card‘𝐴) ⊆ 𝑥))
16 sseq1 3611 . . . . . 6 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ 𝑥𝐴𝑥))
1716imbi2d 330 . . . . 5 ((card‘𝐴) = 𝐴 → ((𝐴𝑥 → (card‘𝐴) ⊆ 𝑥) ↔ (𝐴𝑥𝐴𝑥)))
1815, 17syl5ibcom 235 . . . 4 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = 𝐴 → (𝐴𝑥𝐴𝑥)))
1918ralrimdva 2965 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
20 oncardid 8742 . . . . . . 7 (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21 ensym 7965 . . . . . . 7 ((card‘𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴))
22 endom 7942 . . . . . . 7 (𝐴 ≈ (card‘𝐴) → 𝐴 ≼ (card‘𝐴))
2320, 21, 223syl 18 . . . . . 6 (𝐴 ∈ On → 𝐴 ≼ (card‘𝐴))
2423adantr 481 . . . . 5 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≼ (card‘𝐴))
25 cardon 8730 . . . . . 6 (card‘𝐴) ∈ On
26 breq2 4627 . . . . . . . 8 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ≼ (card‘𝐴)))
27 sseq2 3612 . . . . . . . 8 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ⊆ (card‘𝐴)))
2826, 27imbi12d 334 . . . . . . 7 (𝑥 = (card‘𝐴) → ((𝐴𝑥𝐴𝑥) ↔ (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
2928rspcv 3295 . . . . . 6 ((card‘𝐴) ∈ On → (∀𝑥 ∈ On (𝐴𝑥𝐴𝑥) → (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
3025, 29ax-mp 5 . . . . 5 (∀𝑥 ∈ On (𝐴𝑥𝐴𝑥) → (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴)))
3124, 30syl5com 31 . . . 4 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴𝑥𝐴𝑥) → 𝐴 ⊆ (card‘𝐴)))
32 cardonle 8743 . . . . . . 7 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
3332adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘𝐴) ⊆ 𝐴)
3433biantrurd 529 . . . . 5 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴))))
35 eqss 3603 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
3634, 35syl6bbr 278 . . . 4 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
3731, 36sylibd 229 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴𝑥𝐴𝑥) → (card‘𝐴) = 𝐴))
3819, 37impbid 202 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
394, 38bitrd 268 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wss 3560   class class class wbr 4623  dom cdm 5084  ran crn 5085  Oncon0 5692  cfv 5857  ωcom 7027  cen 7912  cdom 7913  cardccrd 8721  cale 8722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-oi 8375  df-har 8423  df-card 8725  df-aleph 8726
This theorem is referenced by: (None)
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