Theorem alephordi 10125

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 2849	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
2		fveq2 6873	. . . 4 ⊢ (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
3	2	breq2d 5114	. . 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 2849	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
6		fveq2 6873	. . . 4 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
7	6	breq2d 5114	. . 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 2849	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
10		fveq2 6873	. . . 4 ⊢ (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
11	10	breq2d 5114	. . 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 2849	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
14		fveq2 6873	. . . 4 ⊢ (𝑥 = 𝐵 → (ℵ‘𝑥) = (ℵ‘𝐵))
15	14	breq2d 5114	. . 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 4283	. . 3 ⊢ ¬ 𝐴 ∈ ∅
18	17	pm2.21i 120	. 2 ⊢ (𝐴 ∈ ∅ → (ℵ‘𝐴) ≺ (ℵ‘∅))
19		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
20	19	elsuc2 6425	. . . 4 ⊢ (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
21		alephordilem1 10124	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦))
22		sdomtr 9112	. . . . . . . . 9 ⊢ (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ≺ (ℵ‘suc 𝑦)) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
23	21, 22	sylan2 605	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ≺ (ℵ‘𝑦) ∧ 𝑦 ∈ On) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝑦))
24	23	expcom 419	. . . . . . 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 6873	. . . . . . . . 9 ⊢ (𝐴 = 𝑦 → (ℵ‘𝐴) = (ℵ‘𝑦))
28	27	breq1d 5112	. . . . . . . 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 873	. . . 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 6888	. . . . . 6 ⊢ (Lim 𝑥 → (ℵ‘𝑥) ∈ V)
36		fveq2 6873	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (ℵ‘𝑤) = (ℵ‘𝐴))
37	36	ssiun2s 5006	. . . . . . 7 ⊢ (𝐴 ∈ 𝑥 → (ℵ‘𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
38		vex 3454	. . . . . . . . 9 ⊢ 𝑥 ∈ V
39		alephlim 10118	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
40	38, 39	mpan 703	. . . . . . . 8 ⊢ (Lim 𝑥 → (ℵ‘𝑥) = ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
41	40	sseq2d 3962	. . . . . . 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 9005	. . . . . 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 6873	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝐴 → (ℵ‘𝑤) = (ℵ‘suc 𝐴))
47	46	ssiun2s 5006	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → (ℵ‘suc 𝐴) ⊆ ∪ 𝑤 ∈ 𝑥 (ℵ‘𝑤))
48	40	sseq2d 3962	. . . . . . . . . . . 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 9005	. . . . . . . . . . 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 412	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝑥))
54		domnsym 9100	. . . . . . . 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 6417	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
57	38, 56	mpan 703	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
58		onelon 6376	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
59	57, 58	sylan 592	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → 𝐴 ∈ On)
60		ensym 9008	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ≈ (ℵ‘𝑥) → (ℵ‘𝑥) ≈ (ℵ‘𝐴))
61		alephordilem1 10124	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
62		ensdomtr 9110	. . . . . . . . . 10 ⊢ (((ℵ‘𝑥) ≈ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → (ℵ‘𝑥) ≺ (ℵ‘suc 𝐴))
63	62	ex 418	. . . . . . . . 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 418	. . . . 5 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥)))
68	44, 67	jcad 522	. . . 4 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → ((ℵ‘𝐴) ≼ (ℵ‘𝑥) ∧ ¬ (ℵ‘𝐴) ≈ (ℵ‘𝑥))))
69		brsdom 8979	. . . 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 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3898  ∅c0 4278  ∪ ciun 4950   class class class wbr 5102  Oncon0 6351  Lim wlim 6352  suc csuc 6353  ‘cfv 6527   ≈ cen 8948   ≼ cdom 8949   ≺ csdm 8950  ℵcale 9989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-oi 9482  df-har 9529  df-card 9992  df-aleph 9993

This theorem is used by:  alephord  10126  alephval2  10629