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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  ℵcale 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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