Theorem alephordi 8897

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 2690	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		fveq2 6191	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	2	breq2d 4665	. . 3 ⊢ (𝑥 = ∅ → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘∅)))
4	1, 3	imbi12d 334	. 2 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))))
5		eleq2 2690	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		fveq2 6191	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
7	6	breq2d 4665	. . 3 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝑦)))
8	5, 7	imbi12d 334	. 2 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦))))
9		eleq2 2690	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		fveq2 6191	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
11	10	breq2d 4665	. . 3 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
12	9, 11	imbi12d 334	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
13		eleq2 2690	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		fveq2 6191	. . . 4 ⊢ (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
15	14	breq2d 4665	. . 3 ⊢ (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
16	13, 15	imbi12d 334	. 2 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))))
17		noel 3919	. . 3 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 116	. 2 ⊢ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19		vex 3203	. . . . 5 ⊢ 𝑦 ∈ V
20	19	elsuc2 5795	. . . 4 ⊢ (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
21		alephordilem1 8896	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22		sdomtr 8098	. . . . . . . . 9 ⊢ (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
23	21, 22	sylan2 491	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
24	23	expcom 451	. . . . . . 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 6191	. . . . . . . . 9 ⊢ (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
28	27	breq1d 4663	. . . . . . . 8 ⊢ (𝐴 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝑦) ↔ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)))
29	21, 28	syl5ibr 236	. . . . . . 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 395	. . . 4 ⊢ (𝑦 ∈ On → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
33	20, 32	syl5bi 232	. . 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 6203	. . . . . 6 ⊢ (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
36		fveq2 6191	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
37	36	ssiun2s 4564	. . . . . . 7 ⊢ (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
38		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
39		alephlim 8890	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
40	38, 39	mpan 706	. . . . . . . 8 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
41	40	sseq2d 3633	. . . . . . 7 ⊢ (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤)))
42	37, 41	syl5ibr 236	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
43		ssdomg 8001	. . . . . 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 7049	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
46		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
47	46	ssiun2s 4564	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
48	40	sseq2d 3633	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤)))
49	47, 48	syl5ibr 236	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
50		ssdomg 8001	. . . . . . . . . . 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 230	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
53	52	imp 445	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54		domnsym 8086	. . . . . . . 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 5788	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
57	38, 56	mpan 706	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
58		onelon 5748	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
59	57, 58	sylan 488	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
60		ensym 8005	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61		alephordilem1 8896	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62		ensdomtr 8096	. . . . . . . . . 10 ⊢ (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
63	62	ex 450	. . . . . . . . 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 189	. . . . . 6 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))
67	66	ex 450	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
68	44, 67	jcad 555	. . . 4 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69		brsdom 7978	. . . 4 ⊢ ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
70	68, 69	syl6ibr 242	. . 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 7059	1 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520   class class class wbr 4653  Oncon0 5723  Lim wlim 5724  suc csuc 5725  ‘cfv 5888   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  ℵcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  alephord  8898  alephval2  9394