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Theorem ordunisuc2r 4297
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 2617 . . . . . . . . 9 𝑥 ∈ V
21sucid 4211 . . . . . . . 8 𝑥 ∈ suc 𝑥
3 elunii 3635 . . . . . . . 8 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥𝐴) → 𝑥 𝐴)
42, 3mpan 415 . . . . . . 7 (suc 𝑥𝐴𝑥 𝐴)
54imim2i 12 . . . . . 6 ((𝑥𝐴 → suc 𝑥𝐴) → (𝑥𝐴𝑥 𝐴))
65alimi 1387 . . . . 5 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥(𝑥𝐴𝑥 𝐴))
7 df-ral 2360 . . . . 5 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
8 dfss2 3001 . . . . 5 (𝐴 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 𝐴))
96, 7, 83imtr4i 199 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴𝐴 𝐴)
109a1i 9 . . 3 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 𝐴))
11 orduniss 4219 . . 3 (Ord 𝐴 𝐴𝐴)
1210, 11jctird 310 . 2 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴 → (𝐴 𝐴 𝐴𝐴)))
13 eqss 3027 . 2 (𝐴 = 𝐴 ↔ (𝐴 𝐴 𝐴𝐴))
1412, 13syl6ibr 160 1 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1285   = wceq 1287  wcel 1436  wral 2355  wss 2986   cuni 3630  Ord word 4156  suc csuc 4159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-sn 3431  df-uni 3631  df-tr 3905  df-iord 4160  df-suc 4165
This theorem is referenced by: (None)
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