Theorem alephordi 10010

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 2826	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		fveq2 6842	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	2	breq2d 5117	. . 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 2826	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		fveq2 6842	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
7	6	breq2d 5117	. . 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 2826	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		fveq2 6842	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
11	10	breq2d 5117	. . 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 2826	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		fveq2 6842	. . . 4 ⊢ (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
15	14	breq2d 5117	. . 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 4290	. . 3 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 119	. 2 ⊢ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19		vex 3449	. . . . 5 ⊢ 𝑦 ∈ V
20	19	elsuc2 6388	. . . 4 ⊢ (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
21		alephordilem1 10009	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22		sdomtr 9059	. . . . . . . . 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 414	. . . . . . 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 6842	. . . . . . . . 9 ⊢ (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
28	27	breq1d 5115	. . . . . . . 8 ⊢ (𝐴 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝑦) ↔ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)))
29	21, 28	syl5ibr 245	. . . . . . 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 857	. . . 4 ⊢ (𝑦 ∈ On → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
33	20, 32	biimtrid 241	. . 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 6857	. . . . . 6 ⊢ (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
36		fveq2 6842	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
37	36	ssiun2s 5008	. . . . . . 7 ⊢ (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
38		vex 3449	. . . . . . . . 9 ⊢ 𝑥 ∈ V
39		alephlim 10003	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
40	38, 39	mpan 688	. . . . . . . 8 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
41	40	sseq2d 3976	. . . . . . 7 ⊢ (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤)))
42	37, 41	syl5ibr 245	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
43		ssdomg 8940	. . . . . 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 7785	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
46		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
47	46	ssiun2s 5008	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
48	40	sseq2d 3976	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤)))
49	47, 48	syl5ibr 245	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
50		ssdomg 8940	. . . . . . . . . . 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 239	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
53	52	imp 407	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54		domnsym 9043	. . . . . . . 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 6381	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
57	38, 56	mpan 688	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
58		onelon 6342	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
59	57, 58	sylan 580	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
60		ensym 8943	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61		alephordilem1 10009	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62		ensdomtr 9057	. . . . . . . . . 10 ⊢ (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
63	62	ex 413	. . . . . . . . 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 197	. . . . . 6 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))
67	66	ex 413	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
68	44, 67	jcad 513	. . . 4 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69		brsdom 8915	. . . 4 ⊢ ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
70	68, 69	syl6ibr 251	. . 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 7796	1 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  ∪ ciun 4954   class class class wbr 5105  Oncon0 6317  Lim wlim 6318  suc csuc 6319  ‘cfv 6496   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882  ℵcale 9872

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-oi 9446  df-har 9493  df-card 9875  df-aleph 9876

This theorem is referenced by:  alephord  10011  alephval2  10508