MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephordi Structured version   Visualization version   GIF version

Theorem alephordi 9931
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 2825 . . 3 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
2 fveq2 6825 . . . 4 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32breq2d 5104 . . 3 (𝑥 = ∅ → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘∅)))
41, 3imbi12d 344 . 2 (𝑥 = ∅ → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))))
5 eleq2 2825 . . 3 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
6 fveq2 6825 . . . 4 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
76breq2d 5104 . . 3 (𝑥 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝑦)))
85, 7imbi12d 344 . 2 (𝑥 = 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦))))
9 eleq2 2825 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
10 fveq2 6825 . . . 4 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
1110breq2d 5104 . . 3 (𝑥 = suc 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
129, 11imbi12d 344 . 2 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
13 eleq2 2825 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
14 fveq2 6825 . . . 4 (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
1514breq2d 5104 . . 3 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
1613, 15imbi12d 344 . 2 (𝑥 = 𝐵 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))))
17 noel 4277 . . 3 ¬ 𝐴 ∈ ∅
1817pm2.21i 119 . 2 (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19 vex 3445 . . . . 5 𝑦 ∈ V
2019elsuc2 6372 . . . 4 (𝐴 ∈ suc 𝑦 ↔ (𝐴𝑦𝐴 = 𝑦))
21 alephordilem1 9930 . . . . . . . . 9 (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22 sdomtr 8980 . . . . . . . . 9 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2321, 22sylan2 593 . . . . . . . 8 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2423expcom 414 . . . . . . 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 6825 . . . . . . . . 9 (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
2827breq1d 5102 . . . . . . . 8 (𝐴 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝑦) ↔ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)))
2921, 28syl5ibr 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 856 . . . 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 6840 . . . . . 6 (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
36 fveq2 6825 . . . . . . . 8 (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
3736ssiun2s 4995 . . . . . . 7 (𝐴𝑥 → (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
38 vex 3445 . . . . . . . . 9 𝑥 ∈ V
39 alephlim 9924 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4038, 39mpan 687 . . . . . . . 8 (Lim 𝑥 → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4140sseq2d 3964 . . . . . . 7 (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4237, 41syl5ibr 245 . . . . . 6 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
43 ssdomg 8861 . . . . . 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 7763 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
46 fveq2 6825 . . . . . . . . . . . . 13 (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
4746ssiun2s 4995 . . . . . . . . . . . 12 (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
4840sseq2d 3964 . . . . . . . . . . . 12 (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4947, 48syl5ibr 245 . . . . . . . . . . 11 (Lim 𝑥 → (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
50 ssdomg 8861 . . . . . . . . . . 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 407 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54 domnsym 8964 . . . . . . . 8 ((ℵ‘suc 𝐴) ≼ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
5553, 54syl 17 . . . . . . 7 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
56 limelon 6365 . . . . . . . . . 10 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5738, 56mpan 687 . . . . . . . . 9 (Lim 𝑥𝑥 ∈ On)
58 onelon 6327 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴 ∈ On)
5957, 58sylan 580 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → 𝐴 ∈ On)
60 ensym 8864 . . . . . . . . 9 ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61 alephordilem1 9930 . . . . . . . . 9 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62 ensdomtr 8978 . . . . . . . . . 10 (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
6362ex 413 . . . . . . . . 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 413 . . . . 5 (Lim 𝑥 → (𝐴𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
6844, 67jcad 513 . . . 4 (Lim 𝑥 → (𝐴𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69 brsdom 8836 . . . 4 ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
7068, 69syl6ibr 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 7774 1 (𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844   = wceq 1540  wcel 2105  wral 3061  Vcvv 3441  wss 3898  c0 4269   ciun 4941   class class class wbr 5092  Oncon0 6302  Lim wlim 6303  suc csuc 6304  cfv 6479  cen 8801  cdom 8802  csdm 8803  cale 9793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-inf2 9498
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-om 7781  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-oi 9367  df-har 9414  df-card 9796  df-aleph 9797
This theorem is referenced by:  alephord  9932  alephval2  10429
  Copyright terms: Public domain W3C validator