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Theorem dfon2 32921
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 6193 . 2 On = {𝑥 ∣ Ord 𝑥}
2 tz7.7 6215 . . . . . . . . 9 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥)))
3 df-pss 3958 . . . . . . . . 9 (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥))
42, 3syl6bbr 290 . . . . . . . 8 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥𝑦𝑥))
54exbiri 807 . . . . . . 7 (Ord 𝑥 → (Tr 𝑦 → (𝑦𝑥𝑦𝑥)))
65com23 86 . . . . . 6 (Ord 𝑥 → (𝑦𝑥 → (Tr 𝑦𝑦𝑥)))
76impd 411 . . . . 5 (Ord 𝑥 → ((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
87alrimiv 1921 . . . 4 (Ord 𝑥 → ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
9 vex 3503 . . . . . . 7 𝑥 ∈ V
10 dfon2lem3 32914 . . . . . . 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 32918 . . . . . . . 8 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → (𝑡𝑥 → ∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡)))
1413ralrimiv 3186 . . . . . . 7 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → ∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡))
15 dfon2lem9 32920 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → E Fr 𝑥)
16 psseq2 4069 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1716anbi1d 629 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑧 ∧ Tr 𝑢)))
18 elequ2 2122 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1917, 18imbi12d 346 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
2019albidv 1914 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
21 psseq1 4068 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
22 treq 5175 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (Tr 𝑢 ↔ Tr 𝑣))
2321, 22anbi12d 630 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → ((𝑢𝑧 ∧ Tr 𝑢) ↔ (𝑣𝑧 ∧ Tr 𝑣)))
24 elequ1 2114 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
2523, 24imbi12d 346 . . . . . . . . . . . . . 14 (𝑢 = 𝑣 → (((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2625cbvalvw 2036 . . . . . . . . . . . . 13 (∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧))
2720, 26syl6bb 288 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2827rspccv 3624 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑧𝑥 → ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
29 psseq2 4069 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3029anbi1d 629 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑤 ∧ Tr 𝑢)))
31 elequ2 2122 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3230, 31imbi12d 346 . . . . . . . . . . . . . 14 (𝑡 = 𝑤 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
3332albidv 1914 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
34 psseq1 4068 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
35 treq 5175 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (Tr 𝑢 ↔ Tr 𝑦))
3634, 35anbi12d 630 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → ((𝑢𝑤 ∧ Tr 𝑢) ↔ (𝑦𝑤 ∧ Tr 𝑦)))
37 elequ1 2114 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
3836, 37imbi12d 346 . . . . . . . . . . . . . 14 (𝑢 = 𝑦 → (((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
3938cbvalvw 2036 . . . . . . . . . . . . 13 (∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤))
4033, 39syl6bb 288 . . . . . . . . . . . 12 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
4140rspccv 3624 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑤𝑥 → ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
4228, 41anim12d 608 . . . . . . . . . 10 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ((𝑧𝑥𝑤𝑥) → (∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧) ∧ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤))))
43 vex 3503 . . . . . . . . . . 11 𝑧 ∈ V
44 vex 3503 . . . . . . . . . . 11 𝑤 ∈ V
4543, 44dfon2lem5 32916 . . . . . . . . . 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 7484 . . . . . . 7 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)))
51 epel 5468 . . . . . . . . . 10 (𝑧 E 𝑤𝑧𝑤)
52 biid 262 . . . . . . . . . 10 (𝑧 = 𝑤𝑧 = 𝑤)
53 epel 5468 . . . . . . . . . 10 (𝑤 E 𝑧𝑤𝑧)
5451, 52, 533orbi123i 1150 . . . . . . . . 9 ((𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧) ↔ (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
55542ralbii 3171 . . . . . . . 8 (∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧) ↔ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
5655anbi2i 622 . . . . . . 7 (( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)) ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
5750, 56bitri 276 . . . . . 6 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
5849, 57sylibr 235 . . . . 5 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → E We 𝑥)
59 df-ord 6192 . . . . 5 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
6012, 58, 59sylanbrc 583 . . . 4 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Ord 𝑥)
618, 60impbii 210 . . 3 (Ord 𝑥 ↔ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
6261abbii 2891 . 2 {𝑥 ∣ Ord 𝑥} = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
631, 62eqtri 2849 1 On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3o 1080  wal 1528   = wceq 1530  wcel 2107  {cab 2804  wne 3021  wral 3143  Vcvv 3500  wss 3940  wpss 3941   class class class wbr 5063  Tr wtr 5169   E cep 5463   Fr wfr 5510   We wwe 5512  Ord word 6188  Oncon0 6189
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6192  df-on 6193  df-suc 6195
This theorem is referenced by:  dfon3  33237
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