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Theorem alephinit 9509
 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 9506 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
21bicomi 227 . . . 4 (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
32baib 539 . . 3 (ω ⊆ 𝐴 → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
43adantl 485 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
5 onenon 9365 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ dom card)
65adantr 484 . . . . . . 7 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7 onenon 9365 . . . . . . 7 (𝑥 ∈ On → 𝑥 ∈ dom card)
8 carddom2 9393 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝑥 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴𝑥))
96, 7, 8syl2an 598 . . . . . 6 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴𝑥))
10 cardonle 9373 . . . . . . . 8 (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
1110adantl 485 . . . . . . 7 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12 sstr 3923 . . . . . . . 8 (((card‘𝐴) ⊆ (card‘𝑥) ∧ (card‘𝑥) ⊆ 𝑥) → (card‘𝐴) ⊆ 𝑥)
1312expcom 417 . . . . . . 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 263 . . . . 5 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (𝐴𝑥 → (card‘𝐴) ⊆ 𝑥))
16 sseq1 3940 . . . . . 6 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ 𝑥𝐴𝑥))
1716imbi2d 344 . . . . 5 ((card‘𝐴) = 𝐴 → ((𝐴𝑥 → (card‘𝐴) ⊆ 𝑥) ↔ (𝐴𝑥𝐴𝑥)))
1815, 17syl5ibcom 248 . . . 4 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = 𝐴 → (𝐴𝑥𝐴𝑥)))
1918ralrimdva 3154 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
20 oncardid 9372 . . . . . . 7 (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21 ensym 8544 . . . . . . 7 ((card‘𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴))
22 endom 8522 . . . . . . 7 (𝐴 ≈ (card‘𝐴) → 𝐴 ≼ (card‘𝐴))
2320, 21, 223syl 18 . . . . . 6 (𝐴 ∈ On → 𝐴 ≼ (card‘𝐴))
2423adantr 484 . . . . 5 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≼ (card‘𝐴))
25 cardon 9360 . . . . . 6 (card‘𝐴) ∈ On
26 breq2 5035 . . . . . . . 8 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ≼ (card‘𝐴)))
27 sseq2 3941 . . . . . . . 8 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ⊆ (card‘𝐴)))
2826, 27imbi12d 348 . . . . . . 7 (𝑥 = (card‘𝐴) → ((𝐴𝑥𝐴𝑥) ↔ (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
2928rspcv 3566 . . . . . 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 9373 . . . . . . 7 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
3332adantr 484 . . . . . 6 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘𝐴) ⊆ 𝐴)
3433biantrurd 536 . . . . 5 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴))))
35 eqss 3930 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
3634, 35bitr4di 292 . . . 4 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
3731, 36sylibd 242 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴𝑥𝐴𝑥) → (card‘𝐴) = 𝐴))
3819, 37impbid 215 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
394, 38bitrd 282 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881   class class class wbr 5031  dom cdm 5520  ran crn 5521  Oncon0 6160  ‘cfv 6325  ωcom 7563   ≈ cen 8492   ≼ cdom 8493  cardccrd 9351  ℵcale 9352 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-oi 8961  df-har 9008  df-card 9355  df-aleph 9356 This theorem is referenced by: (None)
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