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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((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
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.
StepHypRef Expression
1 df-on 6387 . 2 On = {𝑥 ∣ Ord 𝑥}
2 tz7.7 6409 . . . . . . . . 9 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥)))
3 df-pss 3970 . . . . . . . . 9 (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥))
42, 3bitr4di 289 . . . . . . . 8 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥𝑦𝑥))
54exbiri 810 . . . . . . 7 (Ord 𝑥 → (Tr 𝑦 → (𝑦𝑥𝑦𝑥)))
65com23 86 . . . . . 6 (Ord 𝑥 → (𝑦𝑥 → (Tr 𝑦𝑦𝑥)))
76impd 410 . . . . 5 (Ord 𝑥 → ((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
87alrimiv 1926 . . . 4 (Ord 𝑥 → ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
9 vex 3483 . . . . . . 7 𝑥 ∈ V
10 dfon2lem3 35787 . . . . . . 7 (𝑥 ∈ V → (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → (Tr 𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑧)))
119, 10ax-mp 5 . . . . . 6 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → (Tr 𝑥 ∧ ∀𝑧𝑥 ¬ 𝑧𝑧))
1211simpld 494 . . . . 5 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Tr 𝑥)
139dfon2lem7 35791 . . . . . . . 8 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → (𝑡𝑥 → ∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡)))
1413ralrimiv 3144 . . . . . . 7 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → ∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡))
15 dfon2lem9 35793 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → E Fr 𝑥)
16 psseq2 4090 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1716anbi1d 631 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑧 ∧ Tr 𝑢)))
18 elequ2 2122 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1917, 18imbi12d 344 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
2019albidv 1919 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
21 psseq1 4089 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
22 treq 5266 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (Tr 𝑢 ↔ Tr 𝑣))
2321, 22anbi12d 632 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → ((𝑢𝑧 ∧ Tr 𝑢) ↔ (𝑣𝑧 ∧ Tr 𝑣)))
24 elequ1 2114 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
2523, 24imbi12d 344 . . . . . . . . . . . . . 14 (𝑢 = 𝑣 → (((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2625cbvalvw 2034 . . . . . . . . . . . . 13 (∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧))
2720, 26bitrdi 287 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2827rspccv 3618 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑧𝑥 → ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
29 psseq2 4090 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3029anbi1d 631 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑤 ∧ Tr 𝑢)))
31 elequ2 2122 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3230, 31imbi12d 344 . . . . . . . . . . . . . 14 (𝑡 = 𝑤 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
3332albidv 1919 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
34 psseq1 4089 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
35 treq 5266 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (Tr 𝑢 ↔ Tr 𝑦))
3634, 35anbi12d 632 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → ((𝑢𝑤 ∧ Tr 𝑢) ↔ (𝑦𝑤 ∧ Tr 𝑦)))
37 elequ1 2114 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
3836, 37imbi12d 344 . . . . . . . . . . . . . 14 (𝑢 = 𝑦 → (((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
3938cbvalvw 2034 . . . . . . . . . . . . 13 (∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤))
4033, 39bitrdi 287 . . . . . . . . . . . 12 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
4140rspccv 3618 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑤𝑥 → ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
4228, 41anim12d 609 . . . . . . . . . 10 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ((𝑧𝑥𝑤𝑥) → (∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧) ∧ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤))))
43 vex 3483 . . . . . . . . . . 11 𝑧 ∈ V
44 vex 3483 . . . . . . . . . . 11 𝑤 ∈ V
4543, 44dfon2lem5 35789 . . . . . . . . . 10 ((∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧) ∧ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)) → (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
4642, 45syl6 35 . . . . . . . . 9 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ((𝑧𝑥𝑤𝑥) → (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
4746ralrimivv 3199 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
4815, 47jca 511 . . . . . . 7 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
4914, 48syl 17 . . . . . 6 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
50 dfwe2 7795 . . . . . . 7 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)))
51 epel 5586 . . . . . . . . . 10 (𝑧 E 𝑤𝑧𝑤)
52 biid 261 . . . . . . . . . 10 (𝑧 = 𝑤𝑧 = 𝑤)
53 epel 5586 . . . . . . . . . 10 (𝑤 E 𝑧𝑤𝑧)
5451, 52, 533orbi123i 1156 . . . . . . . . 9 ((𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧) ↔ (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
55542ralbii 3127 . . . . . . . 8 (∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧) ↔ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
5655anbi2i 623 . . . . . . 7 (( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)) ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
5750, 56bitri 275 . . . . . 6 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
5849, 57sylibr 234 . . . . 5 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → E We 𝑥)
59 df-ord 6386 . . . . 5 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
6012, 58, 59sylanbrc 583 . . . 4 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Ord 𝑥)
618, 60impbii 209 . . 3 (Ord 𝑥 ↔ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
6261abbii 2808 . 2 {𝑥 ∣ Ord 𝑥} = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
631, 62eqtri 2764 1 On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
Colors of variables: wff setvar class
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
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