Theorem alephordi 10093

Description: Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
alephordi	⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))

Proof of Theorem alephordi

Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2824	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		fveq2 6881	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	2	breq2d 5136	. . 3 ⊢ (𝑥 = ∅ → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘∅)))
4	1, 3	imbi12d 344	. 2 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))))
5		eleq2 2824	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
7	6	breq2d 5136	. . 3 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝑦)))
8	5, 7	imbi12d 344	. 2 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦))))
9		eleq2 2824	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		fveq2 6881	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
11	10	breq2d 5136	. . 3 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
12	9, 11	imbi12d 344	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
13		eleq2 2824	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
15	14	breq2d 5136	. . 3 ⊢ (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
16	13, 15	imbi12d 344	. 2 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))))
17		noel 4318	. . 3 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 119	. 2 ⊢ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19		vex 3468	. . . . 5 ⊢ 𝑦 ∈ V
20	19	elsuc2 6430	. . . 4 ⊢ (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
21		alephordilem1 10092	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22		sdomtr 9134	. . . . . . . . 9 ⊢ (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
23	21, 22	sylan2 593	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
24	23	expcom 413	. . . . . . 7 ⊢ (𝑦 ∈ On → ((ℵ‘𝐴) ≺ (ℵ‘𝑦) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
25	24	imim2d 57	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
26	25	com23 86	. . . . 5 ⊢ (𝑦 ∈ On → (𝐴 ∈ 𝑦 → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
27		fveq2 6881	. . . . . . . . 9 ⊢ (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
28	27	breq1d 5134	. . . . . . . 8 ⊢ (𝐴 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝑦) ↔ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)))
29	21, 28	imbitrrid 246	. . . . . . 7 ⊢ (𝐴 = 𝑦 → (𝑦 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
30	29	a1d 25	. . . . . 6 ⊢ (𝐴 = 𝑦 → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝑦 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
31	30	com3r 87	. . . . 5 ⊢ (𝑦 ∈ On → (𝐴 = 𝑦 → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
32	26, 31	jaod 859	. . . 4 ⊢ (𝑦 ∈ On → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
33	20, 32	biimtrid 242	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ suc 𝑦 → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
34	33	com23 86	. 2 ⊢ (𝑦 ∈ On → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
35		fvexd 6896	. . . . . 6 ⊢ (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
36		fveq2 6881	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
37	36	ssiun2s 5029	. . . . . . 7 ⊢ (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
38		vex 3468	. . . . . . . . 9 ⊢ 𝑥 ∈ V
39		alephlim 10086	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
40	38, 39	mpan 690	. . . . . . . 8 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
41	40	sseq2d 3996	. . . . . . 7 ⊢ (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤)))
42	37, 41	imbitrrid 246	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
43		ssdomg 9019	. . . . . 6 ⊢ ((ℵ‘𝑥) ∈ V → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) → (ℵ‘𝐴) ≼ (ℵ‘𝑥)))
44	35, 42, 43	sylsyld 61	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≼ (ℵ‘𝑥)))
45		limsuc 7849	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
46		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
47	46	ssiun2s 5029	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
48	40	sseq2d 3996	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤)))
49	47, 48	imbitrrid 246	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
50		ssdomg 9019	. . . . . . . . . . 11 ⊢ ((ℵ‘𝑥) ∈ V → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
51	35, 49, 50	sylsyld 61	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
52	45, 51	sylbid 240	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
53	52	imp 406	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54		domnsym 9118	. . . . . . . 8 ⊢ ((ℵ‘suc 𝐴) ≼ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
55	53, 54	syl 17	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
56		limelon 6422	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
57	38, 56	mpan 690	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
58		onelon 6382	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
59	57, 58	sylan 580	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
60		ensym 9022	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61		alephordilem1 10092	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62		ensdomtr 9132	. . . . . . . . . 10 ⊢ (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
63	62	ex 412	. . . . . . . . 9 ⊢ ((ℵ‘𝑥) ≈ (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
64	60, 61, 63	syl2im 40	. . . . . . . 8 ⊢ ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (𝐴 ∈ On → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
65	59, 64	syl5com 31	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
66	55, 65	mtod 198	. . . . . 6 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))
67	66	ex 412	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
68	44, 67	jcad 512	. . . 4 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69		brsdom 8994	. . . 4 ⊢ ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
70	68, 69	imbitrrdi 252	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)))
71	70	a1d 25	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥))))
72	4, 8, 12, 16, 18, 34, 71	tfinds 7860	1 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  ∪ ciun 4972   class class class wbr 5124  Oncon0 6357  Lim wlim 6358  suc csuc 6359  ‘cfv 6536   ≈ cen 8961   ≼ cdom 8962   ≺ csdm 8963  ℵcale 9955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-oi 9529  df-har 9576  df-card 9958  df-aleph 9959

This theorem is referenced by:  alephord  10094  alephval2  10591