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Theorem alephordi 8935
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 2719 . . 3 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
2 fveq2 6229 . . . 4 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32breq2d 4697 . . 3 (𝑥 = ∅ → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘∅)))
41, 3imbi12d 333 . 2 (𝑥 = ∅ → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))))
5 eleq2 2719 . . 3 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
6 fveq2 6229 . . . 4 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
76breq2d 4697 . . 3 (𝑥 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝑦)))
85, 7imbi12d 333 . 2 (𝑥 = 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦))))
9 eleq2 2719 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
10 fveq2 6229 . . . 4 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
1110breq2d 4697 . . 3 (𝑥 = suc 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
129, 11imbi12d 333 . 2 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
13 eleq2 2719 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
14 fveq2 6229 . . . 4 (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
1514breq2d 4697 . . 3 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
1613, 15imbi12d 333 . 2 (𝑥 = 𝐵 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))))
17 noel 3952 . . 3 ¬ 𝐴 ∈ ∅
1817pm2.21i 116 . 2 (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19 vex 3234 . . . . 5 𝑦 ∈ V
2019elsuc2 5833 . . . 4 (𝐴 ∈ suc 𝑦 ↔ (𝐴𝑦𝐴 = 𝑦))
21 alephordilem1 8934 . . . . . . . . 9 (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22 sdomtr 8139 . . . . . . . . 9 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2321, 22sylan2 490 . . . . . . . 8 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2423expcom 450 . . . . . . 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 6229 . . . . . . . . 9 (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
2827breq1d 4695 . . . . . . . 8 (𝐴 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝑦) ↔ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)))
2921, 28syl5ibr 236 . . . . . . 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 394 . . . 4 (𝑦 ∈ On → ((𝐴𝑦𝐴 = 𝑦) → ((𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
3320, 32syl5bi 232 . . 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 6241 . . . . . 6 (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
36 fveq2 6229 . . . . . . . 8 (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
3736ssiun2s 4596 . . . . . . 7 (𝐴𝑥 → (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
38 vex 3234 . . . . . . . . 9 𝑥 ∈ V
39 alephlim 8928 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4038, 39mpan 706 . . . . . . . 8 (Lim 𝑥 → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4140sseq2d 3666 . . . . . . 7 (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4237, 41syl5ibr 236 . . . . . 6 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
43 ssdomg 8043 . . . . . 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 7091 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
46 fveq2 6229 . . . . . . . . . . . . 13 (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
4746ssiun2s 4596 . . . . . . . . . . . 12 (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
4840sseq2d 3666 . . . . . . . . . . . 12 (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4947, 48syl5ibr 236 . . . . . . . . . . 11 (Lim 𝑥 → (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
50 ssdomg 8043 . . . . . . . . . . 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 230 . . . . . . . . 9 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
5352imp 444 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54 domnsym 8127 . . . . . . . 8 ((ℵ‘suc 𝐴) ≼ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
5553, 54syl 17 . . . . . . 7 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
56 limelon 5826 . . . . . . . . . 10 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5738, 56mpan 706 . . . . . . . . 9 (Lim 𝑥𝑥 ∈ On)
58 onelon 5786 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴 ∈ On)
5957, 58sylan 487 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → 𝐴 ∈ On)
60 ensym 8046 . . . . . . . . 9 ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61 alephordilem1 8934 . . . . . . . . 9 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62 ensdomtr 8137 . . . . . . . . . 10 (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
6362ex 449 . . . . . . . . 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 189 . . . . . 6 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))
6766ex 449 . . . . 5 (Lim 𝑥 → (𝐴𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
6844, 67jcad 554 . . . 4 (Lim 𝑥 → (𝐴𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69 brsdom 8020 . . . 4 ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
7068, 69syl6ibr 242 . . 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 7101 1 (𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948   ciun 4552   class class class wbr 4685  Oncon0 5761  Lim wlim 5762  suc csuc 5763  cfv 5926  cen 7994  cdom 7995  csdm 7996  cale 8800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-oi 8456  df-har 8504  df-card 8803  df-aleph 8804
This theorem is referenced by:  alephord  8936  alephval2  9432
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