ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordunisuc2r GIF version

Theorem ordunisuc2r 4359
Description: An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)
Assertion
Ref Expression
ordunisuc2r (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 = 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ordunisuc2r
StepHypRef Expression
1 vex 2636 . . . . . . . . 9 𝑥 ∈ V
21sucid 4268 . . . . . . . 8 𝑥 ∈ suc 𝑥
3 elunii 3680 . . . . . . . 8 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥𝐴) → 𝑥 𝐴)
42, 3mpan 416 . . . . . . 7 (suc 𝑥𝐴𝑥 𝐴)
54imim2i 12 . . . . . 6 ((𝑥𝐴 → suc 𝑥𝐴) → (𝑥𝐴𝑥 𝐴))
65alimi 1396 . . . . 5 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥(𝑥𝐴𝑥 𝐴))
7 df-ral 2375 . . . . 5 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
8 dfss2 3028 . . . . 5 (𝐴 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 𝐴))
96, 7, 83imtr4i 200 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴𝐴 𝐴)
109a1i 9 . . 3 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 𝐴))
11 orduniss 4276 . . 3 (Ord 𝐴 𝐴𝐴)
1210, 11jctird 311 . 2 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴 → (𝐴 𝐴 𝐴𝐴)))
13 eqss 3054 . 2 (𝐴 = 𝐴 ↔ (𝐴 𝐴 𝐴𝐴))
1412, 13syl6ibr 161 1 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1294   = wceq 1296  wcel 1445  wral 2370  wss 3013   cuni 3675  Ord word 4213  suc csuc 4216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-uni 3676  df-tr 3959  df-iord 4217  df-suc 4222
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator