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Theorem bj-el2oss1o 14797
Description: Shorter proof of el2oss1o 6457 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 6437 . . . 4  |-  1o  e.  On
21ontrci 4439 . . 3  |-  Tr  1o
3 trsucss 4435 . . 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 6431 . 2  |-  2o  =  suc  1o
64, 5eleq2s 2282 1  |-  ( A  e.  2o  ->  A  C_  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2158    C_ wss 3141   Tr wtr 4113   suc csuc 4377   1oc1o 6423   2oc2o 6424
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 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-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445
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-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383  df-1o 6430  df-2o 6431
This theorem is referenced by: (None)
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