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Theorem alephordi 9485
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 2878 . . 3 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
2 fveq2 6645 . . . 4 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32breq2d 5042 . . 3 (𝑥 = ∅ → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘∅)))
41, 3imbi12d 348 . 2 (𝑥 = ∅ → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))))
5 eleq2 2878 . . 3 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
6 fveq2 6645 . . . 4 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
76breq2d 5042 . . 3 (𝑥 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝑦)))
85, 7imbi12d 348 . 2 (𝑥 = 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦))))
9 eleq2 2878 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
10 fveq2 6645 . . . 4 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
1110breq2d 5042 . . 3 (𝑥 = suc 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
129, 11imbi12d 348 . 2 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
13 eleq2 2878 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
14 fveq2 6645 . . . 4 (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
1514breq2d 5042 . . 3 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
1613, 15imbi12d 348 . 2 (𝑥 = 𝐵 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))))
17 noel 4247 . . 3 ¬ 𝐴 ∈ ∅
1817pm2.21i 119 . 2 (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19 vex 3444 . . . . 5 𝑦 ∈ V
2019elsuc2 6229 . . . 4 (𝐴 ∈ suc 𝑦 ↔ (𝐴𝑦𝐴 = 𝑦))
21 alephordilem1 9484 . . . . . . . . 9 (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22 sdomtr 8639 . . . . . . . . 9 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2321, 22sylan2 595 . . . . . . . 8 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2423expcom 417 . . . . . . 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 6645 . . . . . . . . 9 (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
2827breq1d 5040 . . . . . . . 8 (𝐴 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝑦) ↔ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)))
2921, 28syl5ibr 249 . . . . . . 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, 32syl5bi 245 . . 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 6660 . . . . . 6 (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
36 fveq2 6645 . . . . . . . 8 (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
3736ssiun2s 4935 . . . . . . 7 (𝐴𝑥 → (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
38 vex 3444 . . . . . . . . 9 𝑥 ∈ V
39 alephlim 9478 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4038, 39mpan 689 . . . . . . . 8 (Lim 𝑥 → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4140sseq2d 3947 . . . . . . 7 (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4237, 41syl5ibr 249 . . . . . 6 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
43 ssdomg 8538 . . . . . 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 7544 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
46 fveq2 6645 . . . . . . . . . . . . 13 (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
4746ssiun2s 4935 . . . . . . . . . . . 12 (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
4840sseq2d 3947 . . . . . . . . . . . 12 (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4947, 48syl5ibr 249 . . . . . . . . . . 11 (Lim 𝑥 → (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
50 ssdomg 8538 . . . . . . . . . . 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 243 . . . . . . . . 9 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
5352imp 410 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54 domnsym 8627 . . . . . . . 8 ((ℵ‘suc 𝐴) ≼ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
5553, 54syl 17 . . . . . . 7 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
56 limelon 6222 . . . . . . . . . 10 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5738, 56mpan 689 . . . . . . . . 9 (Lim 𝑥𝑥 ∈ On)
58 onelon 6184 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴 ∈ On)
5957, 58sylan 583 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → 𝐴 ∈ On)
60 ensym 8541 . . . . . . . . 9 ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61 alephordilem1 9484 . . . . . . . . 9 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62 ensdomtr 8637 . . . . . . . . . 10 (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
6362ex 416 . . . . . . . . 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 201 . . . . . 6 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))
6766ex 416 . . . . 5 (Lim 𝑥 → (𝐴𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
6844, 67jcad 516 . . . 4 (Lim 𝑥 → (𝐴𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69 brsdom 8515 . . . 4 ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
7068, 69syl6ibr 255 . . 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 7554 1 (𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2111  wral 3106  Vcvv 3441  wss 3881  c0 4243   ciun 4881   class class class wbr 5030  Oncon0 6159  Lim wlim 6160  suc csuc 6161  cfv 6324  cen 8489  cdom 8490  csdm 8491  cale 9349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-oi 8958  df-har 9005  df-card 9352  df-aleph 9353
This theorem is referenced by:  alephord  9486  alephval2  9983
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