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Theorem alephordi 10117
Description: Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephordi (𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))

Proof of Theorem alephordi
Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2815 . . 3 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
2 fveq2 6901 . . . 4 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32breq2d 5165 . . 3 (𝑥 = ∅ → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘∅)))
41, 3imbi12d 343 . 2 (𝑥 = ∅ → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))))
5 eleq2 2815 . . 3 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
6 fveq2 6901 . . . 4 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
76breq2d 5165 . . 3 (𝑥 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝑦)))
85, 7imbi12d 343 . 2 (𝑥 = 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦))))
9 eleq2 2815 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
10 fveq2 6901 . . . 4 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
1110breq2d 5165 . . 3 (𝑥 = suc 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
129, 11imbi12d 343 . 2 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
13 eleq2 2815 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
14 fveq2 6901 . . . 4 (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
1514breq2d 5165 . . 3 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
1613, 15imbi12d 343 . 2 (𝑥 = 𝐵 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))))
17 noel 4333 . . 3 ¬ 𝐴 ∈ ∅
1817pm2.21i 119 . 2 (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19 vex 3466 . . . . 5 𝑦 ∈ V
2019elsuc2 6447 . . . 4 (𝐴 ∈ suc 𝑦 ↔ (𝐴𝑦𝐴 = 𝑦))
21 alephordilem1 10116 . . . . . . . . 9 (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22 sdomtr 9153 . . . . . . . . 9 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2321, 22sylan2 591 . . . . . . . 8 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2423expcom 412 . . . . . . 7 (𝑦 ∈ On → ((ℵ‘𝐴) ≺ (ℵ‘𝑦) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
2524imim2d 57 . . . . . 6 (𝑦 ∈ On → ((𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
2625com23 86 . . . . 5 (𝑦 ∈ On → (𝐴𝑦 → ((𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
27 fveq2 6901 . . . . . . . . 9 (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
2827breq1d 5163 . . . . . . . 8 (𝐴 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝑦) ↔ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)))
2921, 28imbitrrid 245 . . . . . . 7 (𝐴 = 𝑦 → (𝑦 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
3029a1d 25 . . . . . 6 (𝐴 = 𝑦 → ((𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝑦 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
3130com3r 87 . . . . 5 (𝑦 ∈ On → (𝐴 = 𝑦 → ((𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
3226, 31jaod 857 . . . 4 (𝑦 ∈ On → ((𝐴𝑦𝐴 = 𝑦) → ((𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
3320, 32biimtrid 241 . . 3 (𝑦 ∈ On → (𝐴 ∈ suc 𝑦 → ((𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
3433com23 86 . 2 (𝑦 ∈ On → ((𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
35 fvexd 6916 . . . . . 6 (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
36 fveq2 6901 . . . . . . . 8 (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
3736ssiun2s 5056 . . . . . . 7 (𝐴𝑥 → (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
38 vex 3466 . . . . . . . . 9 𝑥 ∈ V
39 alephlim 10110 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4038, 39mpan 688 . . . . . . . 8 (Lim 𝑥 → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4140sseq2d 4012 . . . . . . 7 (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4237, 41imbitrrid 245 . . . . . 6 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
43 ssdomg 9031 . . . . . 6 ((ℵ‘𝑥) ∈ V → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) → (ℵ‘𝐴) ≼ (ℵ‘𝑥)))
4435, 42, 43sylsyld 61 . . . . 5 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ≼ (ℵ‘𝑥)))
45 limsuc 7859 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
46 fveq2 6901 . . . . . . . . . . . . 13 (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
4746ssiun2s 5056 . . . . . . . . . . . 12 (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
4840sseq2d 4012 . . . . . . . . . . . 12 (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4947, 48imbitrrid 245 . . . . . . . . . . 11 (Lim 𝑥 → (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
50 ssdomg 9031 . . . . . . . . . . 11 ((ℵ‘𝑥) ∈ V → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
5135, 49, 50sylsyld 61 . . . . . . . . . 10 (Lim 𝑥 → (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
5245, 51sylbid 239 . . . . . . . . 9 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
5352imp 405 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54 domnsym 9137 . . . . . . . 8 ((ℵ‘suc 𝐴) ≼ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
5553, 54syl 17 . . . . . . 7 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
56 limelon 6440 . . . . . . . . . 10 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5738, 56mpan 688 . . . . . . . . 9 (Lim 𝑥𝑥 ∈ On)
58 onelon 6401 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴 ∈ On)
5957, 58sylan 578 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → 𝐴 ∈ On)
60 ensym 9034 . . . . . . . . 9 ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61 alephordilem1 10116 . . . . . . . . 9 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62 ensdomtr 9151 . . . . . . . . . 10 (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
6362ex 411 . . . . . . . . 9 ((ℵ‘𝑥) ≈ (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
6460, 61, 63syl2im 40 . . . . . . . 8 ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (𝐴 ∈ On → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
6559, 64syl5com 31 . . . . . . 7 ((Lim 𝑥𝐴𝑥) → ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
6655, 65mtod 197 . . . . . 6 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))
6766ex 411 . . . . 5 (Lim 𝑥 → (𝐴𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
6844, 67jcad 511 . . . 4 (Lim 𝑥 → (𝐴𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69 brsdom 9006 . . . 4 ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
7068, 69imbitrrdi 251 . . 3 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)))
7170a1d 25 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥))))
724, 8, 12, 16, 18, 34, 71tfinds 7870 1 (𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462  wss 3947  c0 4325   ciun 5001   class class class wbr 5153  Oncon0 6376  Lim wlim 6377  suc csuc 6378  cfv 6554  cen 8971  cdom 8972  csdm 8973  cale 9979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-oi 9553  df-har 9600  df-card 9982  df-aleph 9983
This theorem is referenced by:  alephord  10118  alephval2  10615
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