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Theorem ordunisuc2r 4211
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 2557 . . . . . . . . 9 𝑥 ∈ V
21sucid 4125 . . . . . . . 8 𝑥 ∈ suc 𝑥
3 elunii 3581 . . . . . . . 8 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥𝐴) → 𝑥 𝐴)
42, 3mpan 400 . . . . . . 7 (suc 𝑥𝐴𝑥 𝐴)
54imim2i 12 . . . . . 6 ((𝑥𝐴 → suc 𝑥𝐴) → (𝑥𝐴𝑥 𝐴))
65alimi 1344 . . . . 5 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥(𝑥𝐴𝑥 𝐴))
7 df-ral 2308 . . . . 5 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
8 dfss2 2931 . . . . 5 (𝐴 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 𝐴))
96, 7, 83imtr4i 190 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴𝐴 𝐴)
109a1i 9 . . 3 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 𝐴))
11 orduniss 4133 . . 3 (Ord 𝐴 𝐴𝐴)
1210, 11jctird 300 . 2 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴 → (𝐴 𝐴 𝐴𝐴)))
13 eqss 2957 . 2 (𝐴 = 𝐴 ↔ (𝐴 𝐴 𝐴𝐴))
1412, 13syl6ibr 151 1 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wal 1241   = wceq 1243  wcel 1393  wral 2303  wss 2914   cuni 3576  Ord word 4070  suc csuc 4073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-sn 3378  df-uni 3577  df-tr 3851  df-iord 4074  df-suc 4079
This theorem is referenced by: (None)
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