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Theorem dfon2 35516
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 6375 . 2 On = {𝑥 ∣ Ord 𝑥}
2 tz7.7 6397 . . . . . . . . 9 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥)))
3 df-pss 3964 . . . . . . . . 9 (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥))
42, 3bitr4di 288 . . . . . . . 8 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥𝑦𝑥))
54exbiri 809 . . . . . . 7 (Ord 𝑥 → (Tr 𝑦 → (𝑦𝑥𝑦𝑥)))
65com23 86 . . . . . 6 (Ord 𝑥 → (𝑦𝑥 → (Tr 𝑦𝑦𝑥)))
76impd 409 . . . . 5 (Ord 𝑥 → ((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
87alrimiv 1922 . . . 4 (Ord 𝑥 → ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
9 vex 3465 . . . . . . 7 𝑥 ∈ V
10 dfon2lem3 35509 . . . . . . 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 493 . . . . 5 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Tr 𝑥)
139dfon2lem7 35513 . . . . . . . 8 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → (𝑡𝑥 → ∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡)))
1413ralrimiv 3134 . . . . . . 7 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → ∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡))
15 dfon2lem9 35515 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → E Fr 𝑥)
16 psseq2 4084 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1716anbi1d 629 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑧 ∧ Tr 𝑢)))
18 elequ2 2113 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1917, 18imbi12d 343 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
2019albidv 1915 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
21 psseq1 4083 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
22 treq 5274 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (Tr 𝑢 ↔ Tr 𝑣))
2321, 22anbi12d 630 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → ((𝑢𝑧 ∧ Tr 𝑢) ↔ (𝑣𝑧 ∧ Tr 𝑣)))
24 elequ1 2105 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
2523, 24imbi12d 343 . . . . . . . . . . . . . 14 (𝑢 = 𝑣 → (((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2625cbvalvw 2031 . . . . . . . . . . . . 13 (∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧))
2720, 26bitrdi 286 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2827rspccv 3603 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑧𝑥 → ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
29 psseq2 4084 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3029anbi1d 629 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑤 ∧ Tr 𝑢)))
31 elequ2 2113 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3230, 31imbi12d 343 . . . . . . . . . . . . . 14 (𝑡 = 𝑤 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
3332albidv 1915 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
34 psseq1 4083 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
35 treq 5274 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (Tr 𝑢 ↔ Tr 𝑦))
3634, 35anbi12d 630 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → ((𝑢𝑤 ∧ Tr 𝑢) ↔ (𝑦𝑤 ∧ Tr 𝑦)))
37 elequ1 2105 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
3836, 37imbi12d 343 . . . . . . . . . . . . . 14 (𝑢 = 𝑦 → (((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
3938cbvalvw 2031 . . . . . . . . . . . . 13 (∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤))
4033, 39bitrdi 286 . . . . . . . . . . . 12 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
4140rspccv 3603 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑤𝑥 → ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
4228, 41anim12d 607 . . . . . . . . . 10 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ((𝑧𝑥𝑤𝑥) → (∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧) ∧ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤))))
43 vex 3465 . . . . . . . . . . 11 𝑧 ∈ V
44 vex 3465 . . . . . . . . . . 11 𝑤 ∈ V
4543, 44dfon2lem5 35511 . . . . . . . . . 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 3188 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
4815, 47jca 510 . . . . . . 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 7777 . . . . . . 7 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)))
51 epel 5585 . . . . . . . . . 10 (𝑧 E 𝑤𝑧𝑤)
52 biid 260 . . . . . . . . . 10 (𝑧 = 𝑤𝑧 = 𝑤)
53 epel 5585 . . . . . . . . . 10 (𝑤 E 𝑧𝑤𝑧)
5451, 52, 533orbi123i 1153 . . . . . . . . 9 ((𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧) ↔ (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
55542ralbii 3117 . . . . . . . 8 (∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧) ↔ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
5655anbi2i 621 . . . . . . 7 (( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)) ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
5750, 56bitri 274 . . . . . 6 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
5849, 57sylibr 233 . . . . 5 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → E We 𝑥)
59 df-ord 6374 . . . . 5 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
6012, 58, 59sylanbrc 581 . . . 4 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Ord 𝑥)
618, 60impbii 208 . . 3 (Ord 𝑥 ↔ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
6261abbii 2795 . 2 {𝑥 ∣ Ord 𝑥} = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
631, 62eqtri 2753 1 On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3o 1083  wal 1531   = wceq 1533  wcel 2098  {cab 2702  wne 2929  wral 3050  Vcvv 3461  wss 3944  wpss 3945   class class class wbr 5149  Tr wtr 5266   E cep 5581   Fr wfr 5630   We wwe 5632  Ord word 6370  Oncon0 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-on 6375  df-suc 6377
This theorem is referenced by:  dfon3  35616
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