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Theorem alephordi 8844
 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 2687 . . 3 (𝑥 = ∅ → (𝐴𝑥𝐴 ∈ ∅))
2 fveq2 6150 . . . 4 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32breq2d 4627 . . 3 (𝑥 = ∅ → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘∅)))
41, 3imbi12d 334 . 2 (𝑥 = ∅ → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))))
5 eleq2 2687 . . 3 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
6 fveq2 6150 . . . 4 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
76breq2d 4627 . . 3 (𝑥 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝑦)))
85, 7imbi12d 334 . 2 (𝑥 = 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦))))
9 eleq2 2687 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑥𝐴 ∈ suc 𝑦))
10 fveq2 6150 . . . 4 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
1110breq2d 4627 . . 3 (𝑥 = suc 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
129, 11imbi12d 334 . 2 (𝑥 = suc 𝑦 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
13 eleq2 2687 . . 3 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
14 fveq2 6150 . . . 4 (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
1514breq2d 4627 . . 3 (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
1613, 15imbi12d 334 . 2 (𝑥 = 𝐵 → ((𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))))
17 noel 3897 . . 3 ¬ 𝐴 ∈ ∅
1817pm2.21i 116 . 2 (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19 vex 3189 . . . . 5 𝑦 ∈ V
2019elsuc2 5756 . . . 4 (𝐴 ∈ suc 𝑦 ↔ (𝐴𝑦𝐴 = 𝑦))
21 alephordilem1 8843 . . . . . . . . 9 (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22 sdomtr 8045 . . . . . . . . 9 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2321, 22sylan2 491 . . . . . . . 8 (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
2423expcom 451 . . . . . . 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 6150 . . . . . . . . 9 (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
2827breq1d 4625 . . . . . . . 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 395 . . . 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 fvex 6160 . . . . . . 7 (ℵ‘𝑥) ∈ V
3635a1i 11 . . . . . 6 (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
37 fveq2 6150 . . . . . . . 8 (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
3837ssiun2s 4532 . . . . . . 7 (𝐴𝑥 → (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
39 vex 3189 . . . . . . . . 9 𝑥 ∈ V
40 alephlim 8837 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4139, 40mpan 705 . . . . . . . 8 (Lim 𝑥 → (ℵ‘𝑥) = 𝑤𝑥 (ℵ‘𝑤))
4241sseq2d 3614 . . . . . . 7 (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
4338, 42syl5ibr 236 . . . . . 6 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
44 ssdomg 7948 . . . . . 6 ((ℵ‘𝑥) ∈ V → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) → (ℵ‘𝐴) ≼ (ℵ‘𝑥)))
4536, 43, 44sylsyld 61 . . . . 5 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ≼ (ℵ‘𝑥)))
46 limsuc 6999 . . . . . . . . . 10 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
47 fveq2 6150 . . . . . . . . . . . . 13 (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
4847ssiun2s 4532 . . . . . . . . . . . 12 (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤))
4941sseq2d 3614 . . . . . . . . . . . 12 (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ 𝑤𝑥 (ℵ‘𝑤)))
5048, 49syl5ibr 236 . . . . . . . . . . 11 (Lim 𝑥 → (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
51 ssdomg 7948 . . . . . . . . . . 11 ((ℵ‘𝑥) ∈ V → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
5236, 50, 51sylsyld 61 . . . . . . . . . 10 (Lim 𝑥 → (suc 𝐴𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
5346, 52sylbid 230 . . . . . . . . 9 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
5453imp 445 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
55 domnsym 8033 . . . . . . . 8 ((ℵ‘suc 𝐴) ≼ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
5654, 55syl 17 . . . . . . 7 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
57 limelon 5749 . . . . . . . . . 10 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5839, 57mpan 705 . . . . . . . . 9 (Lim 𝑥𝑥 ∈ On)
59 onelon 5709 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐴𝑥) → 𝐴 ∈ On)
6058, 59sylan 488 . . . . . . . 8 ((Lim 𝑥𝐴𝑥) → 𝐴 ∈ On)
61 ensym 7952 . . . . . . . . 9 ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
62 alephordilem1 8843 . . . . . . . . 9 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
63 ensdomtr 8043 . . . . . . . . . 10 (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
6463ex 450 . . . . . . . . 9 ((ℵ‘𝑥) ≈ (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
6561, 62, 64syl2im 40 . . . . . . . 8 ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (𝐴 ∈ On → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
6660, 65syl5com 31 . . . . . . 7 ((Lim 𝑥𝐴𝑥) → ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
6756, 66mtod 189 . . . . . 6 ((Lim 𝑥𝐴𝑥) → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))
6867ex 450 . . . . 5 (Lim 𝑥 → (𝐴𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
6945, 68jcad 555 . . . 4 (Lim 𝑥 → (𝐴𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
70 brsdom 7925 . . . 4 ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
7169, 70syl6ibr 242 . . 3 (Lim 𝑥 → (𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)))
7271a1d 25 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝐴𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥))))
734, 8, 12, 16, 18, 34, 72tfinds 7009 1 (𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ⊆ wss 3556  ∅c0 3893  ∪ ciun 4487   class class class wbr 4615  Oncon0 5684  Lim wlim 5685  suc csuc 5686  ‘cfv 5849   ≈ cen 7899   ≼ cdom 7900   ≺ csdm 7901  ℵcale 8709 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-om 7016  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-oi 8362  df-har 8410  df-card 8712  df-aleph 8713 This theorem is referenced by:  alephord  8845  alephval2  9341
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