Theorem dfon2 36295

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 6371	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
2		tz7.7 6393	. . . . . . . . 9 ⊢ ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦 ∈ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≠ 𝑥)))
3		df-pss 3928	. . . . . . . . 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 3462	. . . . . . 7 ⊢ 𝑥 ∈ V
10		dfon2lem3 36288	. . . . . . 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 36292	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → (𝑡 ∈ 𝑥 → ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡)))
14	13	ralrimiv 3159	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → ∀𝑡 ∈ 𝑥 ∀𝑢((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) → 𝑢 ∈ 𝑡))
15		dfon2lem9 36294	. . . . . . . 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 643	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑧 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑧 ∧ Tr 𝑢)))
18		elequ2 2161	. . . . . . . . . . . . . . 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 4047	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑣 → (𝑢 ⊊ 𝑧 ↔ 𝑣 ⊊ 𝑧))
22		treq 5230	. . . . . . . . . . . . . . . 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 2153	. . . . . . . . . . . . . . 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 3581	. . . . . . . . . . 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 643	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑤 → ((𝑢 ⊊ 𝑡 ∧ Tr 𝑢) ↔ (𝑢 ⊊ 𝑤 ∧ Tr 𝑢)))
31		elequ2 2161	. . . . . . . . . . . . . . 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 4047	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑦 → (𝑢 ⊊ 𝑤 ↔ 𝑦 ⊊ 𝑤))
35		treq 5230	. . . . . . . . . . . . . . . 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 2153	. . . . . . . . . . . . . . 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 3581	. . . . . . . . . . 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 3462	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
44		vex 3462	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
45	43, 44	dfon2lem5 36290	. . . . . . . . . 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 3209	. . . . . . . 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 7782	. . . . . . 7 ⊢ ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 E 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 E 𝑧)))
51		epel 5569	. . . . . . . . . 10 ⊢ (𝑧 E 𝑤 ↔ 𝑧 ∈ 𝑤)
52		biid 264	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 ↔ 𝑧 = 𝑤)
53		epel 5569	. . . . . . . . . 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 3143	. . . . . . . 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 6370	. . . . 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 2833	. 2 ⊢ {𝑥 ∣ Ord 𝑥} = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
63	1, 62	eqtri 2789	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 2146  {cab 2744   ≠ wne 2961  ∀wral 3082  Vcvv 3458   ⊆ wss 3908   ⊊ wpss 3909   class class class wbr 5114  Tr wtr 5223   E cep 5565   Fr wfr 5616   We wwe 5618  Ord word 6366  Oncon0 6367

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

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 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6370  df-on 6371  df-suc 6373

This theorem is used by:  dfon3  36395