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Theorem alephinit 8862
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 8859 . . . . 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 8719 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ dom card)
65adantr 481 . . . . . . 7 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7 onenon 8719 . . . . . . 7 (𝑥 ∈ On → 𝑥 ∈ dom card)
8 carddom2 8747 . . . . . . 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 8727 . . . . . . . 8 (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
1110adantl 482 . . . . . . 7 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12 sstr 3591 . . . . . . . 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 3605 . . . . . 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 2963 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
20 oncardid 8726 . . . . . . 7 (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21 ensym 7949 . . . . . . 7 ((card‘𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴))
22 endom 7926 . . . . . . 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 8714 . . . . . 6 (card‘𝐴) ∈ On
26 breq2 4617 . . . . . . . 8 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ≼ (card‘𝐴)))
27 sseq2 3606 . . . . . . . 8 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ⊆ (card‘𝐴)))
2826, 27imbi12d 334 . . . . . . 7 (𝑥 = (card‘𝐴) → ((𝐴𝑥𝐴𝑥) ↔ (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
2928rspcv 3291 . . . . . 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 8727 . . . . . . 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 3598 . . . . 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 2907  wss 3555   class class class wbr 4613  dom cdm 5074  ran crn 5075  Oncon0 5682  cfv 5847  ωcom 7012  cen 7896  cdom 7897  cardccrd 8705  cale 8706
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-oi 8359  df-har 8407  df-card 8709  df-aleph 8710
This theorem is referenced by: (None)
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