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Theorem oneluni 4133
 Description: An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
Hypothesis
Ref Expression
on.1 A On
Assertion
Ref Expression
oneluni (B A → (AB) = A)

Proof of Theorem oneluni
StepHypRef Expression
1 on.1 . . 3 A On
21onelssi 4131 . 2 (B ABA)
3 ssequn2 3110 . 2 (BA ↔ (AB) = A)
42, 3sylib 127 1 (B A → (AB) = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390   ∪ cun 2909   ⊆ wss 2911  Oncon0 4065 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-uni 3571  df-tr 3845  df-iord 4068  df-on 4070 This theorem is referenced by: (None)
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