Theorem dfon2 36534

Description: On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers", American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.)

Assertion

Ref	Expression
dfon2	⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfon2

Dummy variables 𝑧 𝑤 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-on 6365	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
2		tz7.7 6387	. . . . . . . . 9 ⊢ ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦 ∈ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥)))
3		df-pss 3919	. . . . . . . . 9 ⊢ (𝑦 ⊊ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥))
4	2, 3	bitr4di 292	. . . . . . . 8 ⊢ ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦 ∈ 𝑥 ↔ 𝑦 ⊊ 𝑥))
5	4	exbiri 823	. . . . . . 7 ⊢ (Ord 𝑥 → (Tr 𝑦 → (𝑦 ⊊ 𝑥 → 𝑦 ∈ 𝑥)))
6	5	com23 87	. . . . . 6 ⊢ (Ord 𝑥 → (𝑦 ⊊ 𝑥 → (Tr 𝑦 → 𝑦 ∈ 𝑥)))
7	6	impd 416	. . . . 5 ⊢ (Ord 𝑥 → ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
8	7	alrimiv 1960	. . . 4 ⊢ (Ord 𝑥 → ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
9		vex 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
10		dfon2lem3 36527	. . . . . . 7 ⊢ (𝑥 ∈ V → (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → (Tr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)))
11	9, 10	ax-mp 5	. . . . . 6 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → (Tr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧))
12	11	simpld 500	. . . . 5 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → Tr 𝑥)
13	9	dfon2lem7 36531	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → (𝑡 ∈ 𝑥 → ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡)))
14	13	ralrimiv 3154	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → ∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡))
15		dfon2lem9 36533	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → E Fr 𝑥)
16		psseq2 4039	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → (𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑧))
17	16	anbi1d 643	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑧 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑧 ∧ Tr 𝑢)))
18		elequ2 2160	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑧 → (𝑢 ∈ 𝑡 ↔ 𝑢 ∈ 𝑧))
19	17, 18	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → (((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧)))
20	19	albidv 1953	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑢((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧)))
21		psseq1 4038	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑣 → (𝑢 ⊊ 𝑧 ↔ 𝑣 ⊊ 𝑧))
22		treq 5219	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑣 → (Tr 𝑢 ↔ Tr 𝑣))
23	21, 22	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑣 → ((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) ↔ (𝑣 ⊊ 𝑧 ∧ Tr 𝑣)))
24		elequ1 2152	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑣 → (𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧))
25	23, 24	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑣 → (((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧) ↔ ((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
26	25	cbvalvw 2069	. . . . . . . . . . . . 13 ⊢ (∀𝑢((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧) ↔ ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧))
27	20, 26	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
28	27	rspccv 3574	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → (𝑧 ∈ 𝑥 → ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
29		psseq2 4039	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑤 → (𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑤))
30	29	anbi1d 643	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑤 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑤 ∧ Tr 𝑢)))
31		elequ2 2160	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑤 → (𝑢 ∈ 𝑡 ↔ 𝑢 ∈ 𝑤))
32	30, 31	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑤 → (((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤)))
33	32	albidv 1953	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑤 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑢((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤)))
34		psseq1 4038	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → (𝑢 ⊊ 𝑤 ↔ 𝑦 ⊊ 𝑤))
35		treq 5219	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → (Tr 𝑢 ↔ Tr 𝑦))
36	34, 35	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 → ((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) ↔ (𝑦 ⊊ 𝑤 ∧ Tr 𝑦)))
37		elequ1 2152	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 → (𝑢 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
38	36, 37	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑦 → (((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤) ↔ ((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤)))
39	38	cbvalvw 2069	. . . . . . . . . . . . 13 ⊢ (∀𝑢((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤) ↔ ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤))
40	33, 39	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑤 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤)))
41	40	rspccv 3574	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → (𝑤 ∈ 𝑥 → ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤)))
42	28, 41	anim12d 621	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧) ∧ ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤))))
43		vex 3455	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
44		vex 3455	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
45	43, 44	dfon2lem5 36529	. . . . . . . . . 10 ⊢ ((∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧) ∧ ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤)) → (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
46	42, 45	syl6 36	. . . . . . . . 9 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
47	46	ralrimivv 3204	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
48	15, 47	jca 521	. . . . . . 7 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
49	14, 48	syl 18	. . . . . 6 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
50		dfwe2 7786	. . . . . . 7 ⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧)))
51		epel 5554	. . . . . . . . . 10 ⊢ (𝑧 E 𝑤 ↔ 𝑧 ∈ 𝑤)
52		biid 264	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 ↔ 𝑧 = 𝑤)
53		epel 5554	. . . . . . . . . 10 ⊢ (𝑤 E 𝑧 ↔ 𝑤 ∈ 𝑧)
54	51, 52, 53	3orbi123i 1174	. . . . . . . . 9 ⊢ ((𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧) ↔ (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
55	54	2ralbii 3138	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧) ↔ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
56	55	anbi2i 635	. . . . . . 7 ⊢ (( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧)) ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
57	50, 56	bitri 278	. . . . . 6 ⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
58	49, 57	sylibr 237	. . . . 5 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → E We 𝑥)
59		df-ord 6364	. . . . 5 ⊢ (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
60	12, 58, 59	sylanbrc 595	. . . 4 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → Ord 𝑥)
61	8, 60	impbii 212	. . 3 ⊢ (Ord 𝑥 ↔ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
62	61	abbii 2828	. 2 ⊢ {𝑥 ∣ Ord 𝑥} = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
63	1, 62	eqtri 2784	1 ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ w3o 1102  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2739   ≠ wne 2956  ∀wral 3077  Vcvv 3451   ⊆ wss 3899   ⊊ wpss 3900   class class class wbr 5103  Tr wtr 5212   E cep 5550   Fr wfr 5601   We wwe 5603  Ord word 6360  Oncon0 6361

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 2733  ax-sep 5249  ax-pr 5391  ax-un 7749

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-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6364  df-on 6365  df-suc 6367

This theorem is used by:  dfon3  36634