Theorem dfon2 34367

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 6321	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
2		tz7.7 6343	. . . . . . . . 9 ⊢ ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦 ∈ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥)))
3		df-pss 3929	. . . . . . . . 9 ⊢ (𝑦 ⊊ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥))
4	2, 3	bitr4di 288	. . . . . . . 8 ⊢ ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦 ∈ 𝑥 ↔ 𝑦 ⊊ 𝑥))
5	4	exbiri 809	. . . . . . 7 ⊢ (Ord 𝑥 → (Tr 𝑦 → (𝑦 ⊊ 𝑥 → 𝑦 ∈ 𝑥)))
6	5	com23 86	. . . . . 6 ⊢ (Ord 𝑥 → (𝑦 ⊊ 𝑥 → (Tr 𝑦 → 𝑦 ∈ 𝑥)))
7	6	impd 411	. . . . 5 ⊢ (Ord 𝑥 → ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
8	7	alrimiv 1930	. . . 4 ⊢ (Ord 𝑥 → ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
9		vex 3449	. . . . . . 7 ⊢ 𝑥 ∈ V
10		dfon2lem3 34360	. . . . . . 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 495	. . . . 5 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → Tr 𝑥)
13	9	dfon2lem7 34364	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → (𝑡 ∈ 𝑥 → ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡)))
14	13	ralrimiv 3142	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → ∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡))
15		dfon2lem9 34366	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → E Fr 𝑥)
16		psseq2 4048	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → (𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑧))
17	16	anbi1d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑧 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑧 ∧ Tr 𝑢)))
18		elequ2 2121	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑧 → (𝑢 ∈ 𝑡 ↔ 𝑢 ∈ 𝑧))
19	17, 18	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → (((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧)))
20	19	albidv 1923	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑢((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧)))
21		psseq1 4047	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑣 → (𝑢 ⊊ 𝑧 ↔ 𝑣 ⊊ 𝑧))
22		treq 5230	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑣 → (Tr 𝑢 ↔ Tr 𝑣))
23	21, 22	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑣 → ((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) ↔ (𝑣 ⊊ 𝑧 ∧ Tr 𝑣)))
24		elequ1 2113	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑣 → (𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧))
25	23, 24	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑣 → (((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧) ↔ ((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
26	25	cbvalvw 2039	. . . . . . . . . . . . 13 ⊢ (∀𝑢((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧) ↔ ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧))
27	20, 26	bitrdi 286	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
28	27	rspccv 3578	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → (𝑧 ∈ 𝑥 → ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
29		psseq2 4048	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑤 → (𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑤))
30	29	anbi1d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑤 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑤 ∧ Tr 𝑢)))
31		elequ2 2121	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑤 → (𝑢 ∈ 𝑡 ↔ 𝑢 ∈ 𝑤))
32	30, 31	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑤 → (((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤)))
33	32	albidv 1923	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑤 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑢((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤)))
34		psseq1 4047	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → (𝑢 ⊊ 𝑤 ↔ 𝑦 ⊊ 𝑤))
35		treq 5230	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → (Tr 𝑢 ↔ Tr 𝑦))
36	34, 35	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 → ((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) ↔ (𝑦 ⊊ 𝑤 ∧ Tr 𝑦)))
37		elequ1 2113	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 → (𝑢 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
38	36, 37	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑦 → (((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤) ↔ ((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤)))
39	38	cbvalvw 2039	. . . . . . . . . . . . 13 ⊢ (∀𝑢((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤) ↔ ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤))
40	33, 39	bitrdi 286	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑤 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤)))
41	40	rspccv 3578	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → (𝑤 ∈ 𝑥 → ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤)))
42	28, 41	anim12d 609	. . . . . . . . . 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 3449	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
44		vex 3449	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
45	43, 44	dfon2lem5 34362	. . . . . . . . . 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 35	. . . . . . . . 9 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
47	46	ralrimivv 3195	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
48	15, 47	jca 512	. . . . . . 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 17	. . . . . 6 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
50		dfwe2 7708	. . . . . . 7 ⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧)))
51		epel 5540	. . . . . . . . . 10 ⊢ (𝑧 E 𝑤 ↔ 𝑧 ∈ 𝑤)
52		biid 260	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 ↔ 𝑧 = 𝑤)
53		epel 5540	. . . . . . . . . 10 ⊢ (𝑤 E 𝑧 ↔ 𝑤 ∈ 𝑧)
54	51, 52, 53	3orbi123i 1156	. . . . . . . . 9 ⊢ ((𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧) ↔ (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
55	54	2ralbii 3127	. . . . . . . 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 623	. . . . . . 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 274	. . . . . 6 ⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
58	49, 57	sylibr 233	. . . . 5 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → E We 𝑥)
59		df-ord 6320	. . . . 5 ⊢ (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
60	12, 58, 59	sylanbrc 583	. . . 4 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → Ord 𝑥)
61	8, 60	impbii 208	. . 3 ⊢ (Ord 𝑥 ↔ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
62	61	abbii 2806	. 2 ⊢ {𝑥 ∣ Ord 𝑥} = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
63	1, 62	eqtri 2764	1 ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ w3o 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2713   ≠ wne 2943  ∀wral 3064  Vcvv 3445   ⊆ wss 3910   ⊊ wpss 3911   class class class wbr 5105  Tr wtr 5222   E cep 5536   Fr wfr 5585   We wwe 5587  Ord word 6316  Oncon0 6317

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-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

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-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  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-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-tr 5223  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by:  dfon3  34477