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Theorem bj-el2oss1o 15244
Description: Shorter proof of el2oss1o 6491 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-el2oss1o  |-  ( A  e.  2o  ->  A  C_  1o )

Proof of Theorem bj-el2oss1o
StepHypRef Expression
1 1on 6471 . . . 4  |-  1o  e.  On
21ontrci 4456 . . 3  |-  Tr  1o
3 trsucss 4452 . . 3  |-  ( Tr  1o  ->  ( A  e.  suc  1o  ->  A  C_  1o ) )
42, 3ax-mp 5 . 2  |-  ( A  e.  suc  1o  ->  A 
C_  1o )
5 df-2o 6465 . 2  |-  2o  =  suc  1o
64, 5eleq2s 2288 1  |-  ( A  e.  2o  ->  A  C_  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2164    C_ wss 3153   Tr wtr 4127   suc csuc 4394   1oc1o 6457   2oc2o 6458
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4395  df-on 4397  df-suc 4400  df-1o 6464  df-2o 6465
This theorem is referenced by: (None)
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