Theorem dfon2 36647

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 6366	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
2		tz7.7 6388	. . . . . . . . 9 ⊢ ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦 ∈ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥)))
3		df-pss 4014	. . . . . . . . 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 1959	. . . 4 ⊢ (Ord 𝑥 → ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
9		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
10		dfon2lem3 36640	. . . . . . 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 36644	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → (𝑡 ∈ 𝑥 → ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡)))
14	13	ralrimiv 3153	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → ∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡))
15		dfon2lem9 36646	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → E Fr 𝑥)
16		psseq2 4018	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → (𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑧))
17	16	anbi1d 643	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑧 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑧 ∧ Tr 𝑢)))
18		elequ2 2159	. . . . . . . . . . . . . . 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 1952	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑢((𝑢 ⊊ 𝑧 ∧ Tr 𝑢) → 𝑢 ∈ 𝑧)))
21		psseq1 4017	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑣 → (𝑢 ⊊ 𝑧 ↔ 𝑣 ⊊ 𝑧))
22		treq 5218	. . . . . . . . . . . . . . . 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 2151	. . . . . . . . . . . . . . 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 2068	. . . . . . . . . . . . 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 3573	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) → (𝑧 ∈ 𝑥 → ∀𝑣((𝑣 ⊊ 𝑧 ∧ Tr 𝑣) → 𝑣 ∈ 𝑧)))
29		psseq2 4018	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑤 → (𝑢 ⊊ 𝑡 ↔ 𝑢 ⊊ 𝑤))
30	29	anbi1d 643	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑤 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑤 ∧ Tr 𝑢)))
31		elequ2 2159	. . . . . . . . . . . . . . 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 1952	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑤 → (∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡) ↔ ∀𝑢((𝑢 ⊊ 𝑤 ∧ Tr 𝑢) → 𝑢 ∈ 𝑤)))
34		psseq1 4017	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → (𝑢 ⊊ 𝑤 ↔ 𝑦 ⊊ 𝑤))
35		treq 5218	. . . . . . . . . . . . . . . 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 2151	. . . . . . . . . . . . . . 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 2068	. . . . . . . . . . . . 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 3573	. . . . . . . . . . 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 3454	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
44		vex 3454	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
45	43, 44	dfon2lem5 36642	. . . . . . . . . 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 3203	. . . . . . . 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 7789	. . . . . . 7 ⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧)))
51		epel 5553	. . . . . . . . . 10 ⊢ (𝑧 E 𝑤 ↔ 𝑧 ∈ 𝑤)
52		biid 264	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 ↔ 𝑧 = 𝑤)
53		epel 5553	. . . . . . . . . 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 3137	. . . . . . . 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 6365	. . . . 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 2827	. 2 ⊢ {𝑥 ∣ Ord 𝑥} = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
63	1, 62	eqtri 2783	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 1567   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ⊊ wpss 4013   class class class wbr 5102  Tr wtr 5211   E cep 5549   Fr wfr 5600   We wwe 5602  Ord word 6361  Oncon0 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-suc 6368

This theorem is used by:  dfon3  36747