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Theorem alephdom 9495
Description: Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009.)
Assertion
Ref Expression
alephdom ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ≼ (ℵ‘𝐵)))

Proof of Theorem alephdom
StepHypRef Expression
1 onsseleq 6225 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
2 alephord 9489 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
3 sdomdom 8525 . . . . 5 ((ℵ‘𝐴) ≺ (ℵ‘𝐵) → (ℵ‘𝐴) ≼ (ℵ‘𝐵))
42, 3syl6bi 254 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
5 fvex 6676 . . . . . . 7 (ℵ‘𝐴) ∈ V
6 fveq2 6663 . . . . . . 7 (𝐴 = 𝐵 → (ℵ‘𝐴) = (ℵ‘𝐵))
7 eqeng 8531 . . . . . . 7 ((ℵ‘𝐴) ∈ V → ((ℵ‘𝐴) = (ℵ‘𝐵) → (ℵ‘𝐴) ≈ (ℵ‘𝐵)))
85, 6, 7mpsyl 68 . . . . . 6 (𝐴 = 𝐵 → (ℵ‘𝐴) ≈ (ℵ‘𝐵))
98a1i 11 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 → (ℵ‘𝐴) ≈ (ℵ‘𝐵)))
10 endom 8524 . . . . 5 ((ℵ‘𝐴) ≈ (ℵ‘𝐵) → (ℵ‘𝐴) ≼ (ℵ‘𝐵))
119, 10syl6 35 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 → (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
124, 11jaod 853 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵𝐴 = 𝐵) → (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
131, 12sylbid 241 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
14 eloni 6194 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
15 eloni 6194 . . . . . 6 (𝐴 ∈ On → Ord 𝐴)
16 ordtri2or 6279 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐴𝐵))
1714, 15, 16syl2anr 596 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴𝐴𝐵))
1817ord 858 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵𝐴𝐴𝐵))
1918con1d 147 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵𝐵𝐴))
20 alephord 9489 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ (ℵ‘𝐵) ≺ (ℵ‘𝐴)))
2120ancoms 459 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 ↔ (ℵ‘𝐵) ≺ (ℵ‘𝐴)))
22 sdomnen 8526 . . . . 5 ((ℵ‘𝐵) ≺ (ℵ‘𝐴) → ¬ (ℵ‘𝐵) ≈ (ℵ‘𝐴))
23 sdomdom 8525 . . . . . 6 ((ℵ‘𝐵) ≺ (ℵ‘𝐴) → (ℵ‘𝐵) ≼ (ℵ‘𝐴))
24 sbth 8625 . . . . . . 7 (((ℵ‘𝐵) ≼ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≼ (ℵ‘𝐵)) → (ℵ‘𝐵) ≈ (ℵ‘𝐴))
2524ex 413 . . . . . 6 ((ℵ‘𝐵) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) ≼ (ℵ‘𝐵) → (ℵ‘𝐵) ≈ (ℵ‘𝐴)))
2623, 25syl 17 . . . . 5 ((ℵ‘𝐵) ≺ (ℵ‘𝐴) → ((ℵ‘𝐴) ≼ (ℵ‘𝐵) → (ℵ‘𝐵) ≈ (ℵ‘𝐴)))
2722, 26mtod 199 . . . 4 ((ℵ‘𝐵) ≺ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≼ (ℵ‘𝐵))
2821, 27syl6bi 254 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 → ¬ (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
2919, 28syld 47 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵 → ¬ (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
3013, 29impcon4bid 228 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933   class class class wbr 5057  Ord word 6183  Oncon0 6184  cfv 6348  cen 8494  cdom 8495  csdm 8496  cale 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-oi 8962  df-har 9010  df-card 9356  df-aleph 9357
This theorem is referenced by: (None)
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