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Theorem ordunisuc2r 4525
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 2752 . . . . . . . . 9 𝑥 ∈ V
21sucid 4429 . . . . . . . 8 𝑥 ∈ suc 𝑥
3 elunii 3826 . . . . . . . 8 ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥𝐴) → 𝑥 𝐴)
42, 3mpan 424 . . . . . . 7 (suc 𝑥𝐴𝑥 𝐴)
54imim2i 12 . . . . . 6 ((𝑥𝐴 → suc 𝑥𝐴) → (𝑥𝐴𝑥 𝐴))
65alimi 1465 . . . . 5 (∀𝑥(𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥(𝑥𝐴𝑥 𝐴))
7 df-ral 2470 . . . . 5 (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴 → suc 𝑥𝐴))
8 dfss2 3156 . . . . 5 (𝐴 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 𝐴))
96, 7, 83imtr4i 201 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴𝐴 𝐴)
109a1i 9 . . 3 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 𝐴))
11 orduniss 4437 . . 3 (Ord 𝐴 𝐴𝐴)
1210, 11jctird 317 . 2 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴 → (𝐴 𝐴 𝐴𝐴)))
13 eqss 3182 . 2 (𝐴 = 𝐴 ↔ (𝐴 𝐴 𝐴𝐴))
1412, 13imbitrrdi 162 1 (Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1361   = wceq 1363  wcel 2158  wral 2465  wss 3141   cuni 3821  Ord word 4374  suc csuc 4377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-uni 3822  df-tr 4114  df-iord 4378  df-suc 4383
This theorem is referenced by: (None)
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