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Theorem dfon2 34367
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 6321 . 2 On = {𝑥 ∣ Ord 𝑥}
2 tz7.7 6343 . . . . . . . . 9 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥)))
3 df-pss 3929 . . . . . . . . 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 411 . . . . 5 (Ord 𝑥 → ((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
87alrimiv 1930 . . . 4 (Ord 𝑥 → ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
9 vex 3449 . . . . . . 7 𝑥 ∈ V
10 dfon2lem3 34360 . . . . . . 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 495 . . . . 5 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Tr 𝑥)
139dfon2lem7 34364 . . . . . . . 8 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → (𝑡𝑥 → ∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡)))
1413ralrimiv 3142 . . . . . . 7 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → ∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡))
15 dfon2lem9 34366 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → E Fr 𝑥)
16 psseq2 4048 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1716anbi1d 630 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑧 ∧ Tr 𝑢)))
18 elequ2 2121 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1917, 18imbi12d 344 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
2019albidv 1923 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
21 psseq1 4047 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
22 treq 5230 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (Tr 𝑢 ↔ Tr 𝑣))
2321, 22anbi12d 631 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → ((𝑢𝑧 ∧ Tr 𝑢) ↔ (𝑣𝑧 ∧ Tr 𝑣)))
24 elequ1 2113 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
2523, 24imbi12d 344 . . . . . . . . . . . . . 14 (𝑢 = 𝑣 → (((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2625cbvalvw 2039 . . . . . . . . . . . . 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 3578 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑧𝑥 → ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
29 psseq2 4048 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3029anbi1d 630 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑤 ∧ Tr 𝑢)))
31 elequ2 2121 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3230, 31imbi12d 344 . . . . . . . . . . . . . 14 (𝑡 = 𝑤 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
3332albidv 1923 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
34 psseq1 4047 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
35 treq 5230 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (Tr 𝑢 ↔ Tr 𝑦))
3634, 35anbi12d 631 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → ((𝑢𝑤 ∧ Tr 𝑢) ↔ (𝑦𝑤 ∧ Tr 𝑦)))
37 elequ1 2113 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
3836, 37imbi12d 344 . . . . . . . . . . . . . 14 (𝑢 = 𝑦 → (((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
3938cbvalvw 2039 . . . . . . . . . . . . 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 3578 . . . . . . . . . . 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 3449 . . . . . . . . . . 11 𝑧 ∈ V
44 vex 3449 . . . . . . . . . . 11 𝑤 ∈ V
4543, 44dfon2lem5 34362 . . . . . . . . . 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 3195 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
4815, 47jca 512 . . . . . . 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 7708 . . . . . . 7 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)))
51 epel 5540 . . . . . . . . . 10 (𝑧 E 𝑤𝑧𝑤)
52 biid 260 . . . . . . . . . 10 (𝑧 = 𝑤𝑧 = 𝑤)
53 epel 5540 . . . . . . . . . 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 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 6320 . . . . 5 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
6012, 58, 59sylanbrc 583 . . . 4 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Ord 𝑥)
618, 60impbii 208 . . 3 (Ord 𝑥 ↔ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
6261abbii 2806 . 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 396  w3o 1086  wal 1539   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  Vcvv 3445  wss 3910  wpss 3911   class class class wbr 5105  Tr wtr 5222   E cep 5536   Fr wfr 5585   We wwe 5587  Ord word 6316  Oncon0 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-tr 5223  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-ord 6320  df-on 6321  df-suc 6323
This theorem is referenced by:  dfon3  34477
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