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Theorem dfon2 32072
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 5912 . 2 On = {𝑥 ∣ Ord 𝑥}
2 tz7.7 5934 . . . . . . . . 9 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥)))
3 df-pss 3748 . . . . . . . . 9 (𝑦𝑥 ↔ (𝑦𝑥𝑦𝑥))
42, 3syl6bbr 280 . . . . . . . 8 ((Ord 𝑥 ∧ Tr 𝑦) → (𝑦𝑥𝑦𝑥))
54exbiri 845 . . . . . . 7 (Ord 𝑥 → (Tr 𝑦 → (𝑦𝑥𝑦𝑥)))
65com23 86 . . . . . 6 (Ord 𝑥 → (𝑦𝑥 → (Tr 𝑦𝑦𝑥)))
76impd 398 . . . . 5 (Ord 𝑥 → ((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
87alrimiv 2022 . . . 4 (Ord 𝑥 → ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
9 vex 3353 . . . . . . 7 𝑥 ∈ V
10 dfon2lem3 32065 . . . . . . 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 488 . . . . 5 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Tr 𝑥)
139dfon2lem7 32069 . . . . . . . 8 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → (𝑡𝑥 → ∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡)))
1413ralrimiv 3112 . . . . . . 7 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → ∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡))
15 dfon2lem9 32071 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → E Fr 𝑥)
16 psseq2 3856 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1716anbi1d 623 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑧 ∧ Tr 𝑢)))
18 elequ2 2169 . . . . . . . . . . . . . . 15 (𝑡 = 𝑧 → (𝑢𝑡𝑢𝑧))
1917, 18imbi12d 335 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
2019albidv 2015 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧)))
21 psseq1 3855 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
22 treq 4917 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑣 → (Tr 𝑢 ↔ Tr 𝑣))
2321, 22anbi12d 624 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → ((𝑢𝑧 ∧ Tr 𝑢) ↔ (𝑣𝑧 ∧ Tr 𝑣)))
24 elequ1 2162 . . . . . . . . . . . . . . 15 (𝑢 = 𝑣 → (𝑢𝑧𝑣𝑧))
2523, 24imbi12d 335 . . . . . . . . . . . . . 14 (𝑢 = 𝑣 → (((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2625cbvalvw 2136 . . . . . . . . . . . . 13 (∀𝑢((𝑢𝑧 ∧ Tr 𝑢) → 𝑢𝑧) ↔ ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧))
2720, 26syl6bb 278 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
2827rspccv 3458 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑧𝑥 → ∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧)))
29 psseq2 3856 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3029anbi1d 623 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → ((𝑢𝑡 ∧ Tr 𝑢) ↔ (𝑢𝑤 ∧ Tr 𝑢)))
31 elequ2 2169 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → (𝑢𝑡𝑢𝑤))
3230, 31imbi12d 335 . . . . . . . . . . . . . 14 (𝑡 = 𝑤 → (((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
3332albidv 2015 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤)))
34 psseq1 3855 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
35 treq 4917 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑦 → (Tr 𝑢 ↔ Tr 𝑦))
3634, 35anbi12d 624 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → ((𝑢𝑤 ∧ Tr 𝑢) ↔ (𝑦𝑤 ∧ Tr 𝑦)))
37 elequ1 2162 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦 → (𝑢𝑤𝑦𝑤))
3836, 37imbi12d 335 . . . . . . . . . . . . . 14 (𝑢 = 𝑦 → (((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
3938cbvalvw 2136 . . . . . . . . . . . . 13 (∀𝑢((𝑢𝑤 ∧ Tr 𝑢) → 𝑢𝑤) ↔ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤))
4033, 39syl6bb 278 . . . . . . . . . . . 12 (𝑡 = 𝑤 → (∀𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) ↔ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
4140rspccv 3458 . . . . . . . . . . 11 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → (𝑤𝑥 → ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤)))
4228, 41anim12d 602 . . . . . . . . . 10 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ((𝑧𝑥𝑤𝑥) → (∀𝑣((𝑣𝑧 ∧ Tr 𝑣) → 𝑣𝑧) ∧ ∀𝑦((𝑦𝑤 ∧ Tr 𝑦) → 𝑦𝑤))))
43 vex 3353 . . . . . . . . . . 11 𝑧 ∈ V
44 vex 3353 . . . . . . . . . . 11 𝑤 ∈ V
4543, 44dfon2lem5 32067 . . . . . . . . . 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 3117 . . . . . . . 8 (∀𝑡𝑥𝑢((𝑢𝑡 ∧ Tr 𝑢) → 𝑢𝑡) → ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
4815, 47jca 507 . . . . . . 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 7179 . . . . . . 7 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)))
51 epel 5193 . . . . . . . . . 10 (𝑧 E 𝑤𝑧𝑤)
52 biid 252 . . . . . . . . . 10 (𝑧 = 𝑤𝑧 = 𝑤)
53 epel 5193 . . . . . . . . . 10 (𝑤 E 𝑧𝑤𝑧)
5451, 52, 533orbi123i 1195 . . . . . . . . 9 ((𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧) ↔ (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
55542ralbii 3128 . . . . . . . 8 (∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧) ↔ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧))
5655anbi2i 616 . . . . . . 7 (( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧 E 𝑤𝑧 = 𝑤𝑤 E 𝑧)) ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
5750, 56bitri 266 . . . . . 6 ( E We 𝑥 ↔ ( E Fr 𝑥 ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤𝑧 = 𝑤𝑤𝑧)))
5849, 57sylibr 225 . . . . 5 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → E We 𝑥)
59 df-ord 5911 . . . . 5 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
6012, 58, 59sylanbrc 578 . . . 4 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → Ord 𝑥)
618, 60impbii 200 . . 3 (Ord 𝑥 ↔ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
6261abbii 2882 . 2 {𝑥 ∣ Ord 𝑥} = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
631, 62eqtri 2787 1 On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3o 1106  wal 1650   = wceq 1652  wcel 2155  {cab 2751  wne 2937  wral 3055  Vcvv 3350  wss 3732  wpss 3733   class class class wbr 4809  Tr wtr 4911   E cep 5189   Fr wfr 5233   We wwe 5235  Ord word 5907  Oncon0 5908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912  df-suc 5914
This theorem is referenced by:  dfon3  32375
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