Theorem alephordi 10065

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 2851	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		fveq2 6881	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	2	breq2d 5120	. . 3 ⊢ (𝑥 = ∅ → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘∅)))
4	1, 3	imbi12d 347	. 2 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))))
5		eleq2 2851	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
7	6	breq2d 5120	. . 3 ⊢ (𝑥 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝑦)))
8	5, 7	imbi12d 347	. 2 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦))))
9		eleq2 2851	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		fveq2 6881	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
11	10	breq2d 5120	. . 3 ⊢ (𝑥 = suc 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
12	9, 11	imbi12d 347	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
13		eleq2 2851	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
15	14	breq2d 5120	. . 3 ⊢ (𝑥 = 𝐵 → ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
16	13, 15	imbi12d 347	. 2 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)) ↔ (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))))
17		noel 4290	. . 3 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 120	. 2 ⊢ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
20	19	elsuc2 6434	. . . 4 ⊢ (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
21		alephordilem1 10064	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22		sdomtr 9101	. . . . . . . . 9 ⊢ (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
23	21, 22	sylan2 604	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
24	23	expcom 418	. . . . . . 7 ⊢ (𝑦 ∈ On → ((ℵ‘𝐴) ≺ (ℵ‘𝑦) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
25	24	imim2d 58	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
26	25	com23 87	. . . . 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 5118	. . . . . . . 8 ⊢ (𝐴 = 𝑦 → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝑦) ↔ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)))
29	21, 28	imbitrrid 249	. . . . . . 7 ⊢ (𝐴 = 𝑦 → (𝑦 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦)))
30	29	a1d 26	. . . . . 6 ⊢ (𝐴 = 𝑦 → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (𝑦 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
31	30	com3r 88	. . . . 5 ⊢ (𝑦 ∈ On → (𝐴 = 𝑦 → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
32	26, 31	jaod 872	. . . 4 ⊢ (𝑦 ∈ On → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
33	20, 32	biimtrid 245	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ suc 𝑦 → ((𝐴 ∈ 𝑦 → (ℵ‘𝐴) ≺ (ℵ‘𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))))
34	33	com23 87	. 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 5012	. . . . . . 7 ⊢ (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
38		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
39		alephlim 10058	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
40	38, 39	mpan 702	. . . . . . . 8 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
41	40	sseq2d 3968	. . . . . . 7 ⊢ (Lim 𝑥 → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤)))
42	37, 41	imbitrrid 249	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ (ℵ‘𝑥)))
43		ssdomg 8995	. . . . . 6 ⊢ ((ℵ‘𝑥) ∈ V → ((ℵ‘𝐴) ⊆ (ℵ‘𝑥) → (ℵ‘𝐴) ≼ (ℵ‘𝑥)))
44	35, 42, 43	sylsyld 62	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≼ (ℵ‘𝑥)))
45		limsuc 7843	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
46		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
47	46	ssiun2s 5012	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
48	40	sseq2d 3968	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) ↔ (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤)))
49	47, 48	imbitrrid 249	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥)))
50		ssdomg 8995	. . . . . . . . . . 11 ⊢ ((ℵ‘𝑥) ∈ V → ((ℵ‘suc 𝐴) ⊆ (ℵ‘𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
51	35, 49, 50	sylsyld 62	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
52	45, 51	sylbid 243	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥)))
53	52	imp 411	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54		domnsym 9089	. . . . . . . 8 ⊢ ((ℵ‘suc 𝐴) ≼ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
55	53, 54	syl 18	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ¬ (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
56		limelon 6426	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
57	38, 56	mpan 702	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
58		onelon 6385	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
59	57, 58	sylan 591	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
60		ensym 8998	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61		alephordilem1 10064	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62		ensdomtr 9099	. . . . . . . . . 10 ⊢ (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
63	62	ex 417	. . . . . . . . 9 ⊢ ((ℵ‘𝑥) ≈ (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
64	60, 61, 63	syl2im 41	. . . . . . . 8 ⊢ ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (𝐴 ∈ On → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
65	59, 64	syl5com 32	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴)))
66	55, 65	mtod 201	. . . . . 6 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))
67	66	ex 417	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
68	44, 67	jcad 521	. . . 4 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69		brsdom 8969	. . . 4 ⊢ ((ℵ‘𝐴) ≺ (ℵ‘𝑥) ↔ ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
70	68, 69	imbitrrdi 255	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ≺ (ℵ‘𝑥)))
71	70	a1d 26	. 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 7854	1 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  ∪ ciun 4955   class class class wbr 5108  Oncon0 6360  Lim wlim 6361  suc csuc 6362  ‘cfv 6536   ≈ cen 8938   ≼ cdom 8939   ≺ csdm 8940  ℵcale 9929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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 7369  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-er 8692  df-en 8942  df-dom 8943  df-sdom 8944  df-oi 9470  df-har 9517  df-card 9932  df-aleph 9933

This theorem is used by:  alephord  10066  alephval2  10563