Theorem dfon2 35794

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 6387	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
2		tz7.7 6409	. . . . . . . . 9 ⊢ ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦 ∈ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥)))
3		df-pss 3970	. . . . . . . . 9 ⊢ (𝑦 ⊊ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥))
4	2, 3	bitr4di 289	. . . . . . . 8 ⊢ ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦 ∈ 𝑥 ↔ 𝑦 ⊊ 𝑥))
5	4	exbiri 810	. . . . . . 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 410	. . . . 5 ⊢ (Ord 𝑥 → ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
8	7	alrimiv 1926	. . . 4 ⊢ (Ord 𝑥 → ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
9		vex 3483	. . . . . . 7 ⊢ 𝑥 ∈ V
10		dfon2lem3 35787	. . . . . . 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 494	. . . . 5 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → Tr 𝑥)
13	9	dfon2lem7 35791	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → (𝑡 ∈ 𝑥 → ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡)))
14	13	ralrimiv 3144	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → ∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡))
15		dfon2lem9 35793	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → E Fr 𝑥)
16		psseq2 4090	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → (𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑧))
17	16	anbi1d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑧 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑧 ∧ Tr 𝑢)))
18		elequ2 2122	. . . . . . . . . . . . . . 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 1919	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑢((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧)))
21		psseq1 4089	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑣 → (𝑢 ⊊ 𝑧 ↔ 𝑣 ⊊ 𝑧))
22		treq 5266	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑣 → (Tr 𝑢 ↔ Tr 𝑣))
23	21, 22	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑣 → ((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) ↔ (𝑣 ⊊ 𝑧 ∧ Tr 𝑣)))
24		elequ1 2114	. . . . . . . . . . . . . . 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 2034	. . . . . . . . . . . . 13 ⊢ (∀𝑢((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧) ↔ ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧))
27	20, 26	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
28	27	rspccv 3618	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → (𝑧 ∈ 𝑥 → ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
29		psseq2 4090	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑤 → (𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑤))
30	29	anbi1d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑤 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑤 ∧ Tr 𝑢)))
31		elequ2 2122	. . . . . . . . . . . . . . 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 1919	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑤 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑢((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤)))
34		psseq1 4089	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → (𝑢 ⊊ 𝑤 ↔ 𝑦 ⊊ 𝑤))
35		treq 5266	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → (Tr 𝑢 ↔ Tr 𝑦))
36	34, 35	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 → ((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) ↔ (𝑦 ⊊ 𝑤 ∧ Tr 𝑦)))
37		elequ1 2114	. . . . . . . . . . . . . . 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 2034	. . . . . . . . . . . . 13 ⊢ (∀𝑢((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤) ↔ ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤))
40	33, 39	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑤 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑦((𝑦 ⊊ 𝑤 ∧ Tr 𝑦) → 𝑦 ∈ 𝑤)))
41	40	rspccv 3618	. . . . . . . . . . 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 3483	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
44		vex 3483	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
45	43, 44	dfon2lem5 35789	. . . . . . . . . 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 3199	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
48	15, 47	jca 511	. . . . . . 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 7795	. . . . . . 7 ⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧)))
51		epel 5586	. . . . . . . . . 10 ⊢ (𝑧 E 𝑤 ↔ 𝑧 ∈ 𝑤)
52		biid 261	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 ↔ 𝑧 = 𝑤)
53		epel 5586	. . . . . . . . . 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 275	. . . . . 6 ⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧)))
58	49, 57	sylibr 234	. . . . 5 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → E We 𝑥)
59		df-ord 6386	. . . . 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 209	. . 3 ⊢ (Ord 𝑥 ↔ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
62	61	abbii 2808	. 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 395   ∨ w3o 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2107  {cab 2713   ≠ wne 2939  ∀wral 3060  Vcvv 3479   ⊆ wss 3950   ⊊ wpss 3951   class class class wbr 5142  Tr wtr 5258   E cep 5582   Fr wfr 5633   We wwe 5635  Ord word 6382  Oncon0 6383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-suc 6389

This theorem is referenced by:  dfon3  35894