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Theorem alephdom 9094
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 5926 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
2 alephord 9088 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
3 sdomdom 8149 . . . . 5 ((ℵ‘𝐴) ≺ (ℵ‘𝐵) → (ℵ‘𝐴) ≼ (ℵ‘𝐵))
42, 3syl6bi 243 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
5 fvex 6362 . . . . . . 7 (ℵ‘𝐴) ∈ V
6 fveq2 6352 . . . . . . 7 (𝐴 = 𝐵 → (ℵ‘𝐴) = (ℵ‘𝐵))
7 eqeng 8155 . . . . . . 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 8148 . . . . 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 394 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵𝐴 = 𝐵) → (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
131, 12sylbid 230 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
14 eloni 5894 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
15 eloni 5894 . . . . . 6 (𝐴 ∈ On → Ord 𝐴)
16 ordtri2or 5983 . . . . . 6 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐴𝐵))
1714, 15, 16syl2anr 496 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴𝐴𝐵))
1817ord 391 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵𝐴𝐴𝐵))
1918con1d 139 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴𝐵𝐵𝐴))
20 alephord 9088 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ (ℵ‘𝐵) ≺ (ℵ‘𝐴)))
2120ancoms 468 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 ↔ (ℵ‘𝐵) ≺ (ℵ‘𝐴)))
22 sdomnen 8150 . . . . 5 ((ℵ‘𝐵) ≺ (ℵ‘𝐴) → ¬ (ℵ‘𝐵) ≈ (ℵ‘𝐴))
23 sdomdom 8149 . . . . . 6 ((ℵ‘𝐵) ≺ (ℵ‘𝐴) → (ℵ‘𝐵) ≼ (ℵ‘𝐴))
24 sbth 8245 . . . . . . 7 (((ℵ‘𝐵) ≼ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≼ (ℵ‘𝐵)) → (ℵ‘𝐵) ≈ (ℵ‘𝐴))
2524ex 449 . . . . . 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 189 . . . 4 ((ℵ‘𝐵) ≺ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≼ (ℵ‘𝐵))
2821, 27syl6bi 243 . . 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 217 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ≼ (ℵ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715   class class class wbr 4804  Ord word 5883  Oncon0 5884  cfv 6049  cen 8118  cdom 8119  csdm 8120  cale 8952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-oi 8580  df-har 8628  df-card 8955  df-aleph 8956
This theorem is referenced by: (None)
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