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Theorem alephinit 10164
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 10161 . . . . 5 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
21bicomi 224 . . . 4 (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
32baib 535 . . 3 (ω ⊆ 𝐴 → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
43adantl 481 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ (card‘𝐴) = 𝐴))
5 onenon 10018 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ dom card)
65adantr 480 . . . . . . 7 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ∈ dom card)
7 onenon 10018 . . . . . . 7 (𝑥 ∈ On → 𝑥 ∈ dom card)
8 carddom2 10046 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝑥 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴𝑥))
96, 7, 8syl2an 595 . . . . . 6 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) ⊆ (card‘𝑥) ↔ 𝐴𝑥))
10 cardonle 10026 . . . . . . . 8 (𝑥 ∈ On → (card‘𝑥) ⊆ 𝑥)
1110adantl 481 . . . . . . 7 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (card‘𝑥) ⊆ 𝑥)
12 sstr 4017 . . . . . . . 8 (((card‘𝐴) ⊆ (card‘𝑥) ∧ (card‘𝑥) ⊆ 𝑥) → (card‘𝐴) ⊆ 𝑥)
1312expcom 413 . . . . . . 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 260 . . . . 5 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → (𝐴𝑥 → (card‘𝐴) ⊆ 𝑥))
16 sseq1 4034 . . . . . 6 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ 𝑥𝐴𝑥))
1716imbi2d 340 . . . . 5 ((card‘𝐴) = 𝐴 → ((𝐴𝑥 → (card‘𝐴) ⊆ 𝑥) ↔ (𝐴𝑥𝐴𝑥)))
1815, 17syl5ibcom 245 . . . 4 (((𝐴 ∈ On ∧ ω ⊆ 𝐴) ∧ 𝑥 ∈ On) → ((card‘𝐴) = 𝐴 → (𝐴𝑥𝐴𝑥)))
1918ralrimdva 3160 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 → ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
20 oncardid 10025 . . . . . . 7 (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
21 ensym 9063 . . . . . . 7 ((card‘𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴))
22 endom 9039 . . . . . . 7 (𝐴 ≈ (card‘𝐴) → 𝐴 ≼ (card‘𝐴))
2320, 21, 223syl 18 . . . . . 6 (𝐴 ∈ On → 𝐴 ≼ (card‘𝐴))
2423adantr 480 . . . . 5 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≼ (card‘𝐴))
25 cardon 10013 . . . . . 6 (card‘𝐴) ∈ On
26 breq2 5170 . . . . . . . 8 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ≼ (card‘𝐴)))
27 sseq2 4035 . . . . . . . 8 (𝑥 = (card‘𝐴) → (𝐴𝑥𝐴 ⊆ (card‘𝐴)))
2826, 27imbi12d 344 . . . . . . 7 (𝑥 = (card‘𝐴) → ((𝐴𝑥𝐴𝑥) ↔ (𝐴 ≼ (card‘𝐴) → 𝐴 ⊆ (card‘𝐴))))
2928rspcv 3631 . . . . . 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 10026 . . . . . . 7 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
3332adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘𝐴) ⊆ 𝐴)
3433biantrurd 532 . . . . 5 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴))))
35 eqss 4024 . . . . 5 ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴𝐴 ⊆ (card‘𝐴)))
3634, 35bitr4di 289 . . . 4 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴))
3731, 36sylibd 239 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (∀𝑥 ∈ On (𝐴𝑥𝐴𝑥) → (card‘𝐴) = 𝐴))
3819, 37impbid 212 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
394, 38bitrd 279 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  wral 3067  wss 3976   class class class wbr 5166  dom cdm 5700  ran crn 5701  Oncon0 6395  cfv 6573  ωcom 7903  cen 9000  cdom 9001  cardccrd 10004  cale 10005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-har 9626  df-card 10008  df-aleph 10009
This theorem is referenced by: (None)
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