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Theorem bj-el2oss1o 13152
Description: Shorter proof of el2oss1o 13359 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-el2oss1o (𝐴 ∈ 2o𝐴 ⊆ 1o)

Proof of Theorem bj-el2oss1o
StepHypRef Expression
1 1on 6328 . . . 4 1o ∈ On
21ontrci 4357 . . 3 Tr 1o
3 trsucss 4353 . . 3 (Tr 1o → (𝐴 ∈ suc 1o𝐴 ⊆ 1o))
42, 3ax-mp 5 . 2 (𝐴 ∈ suc 1o𝐴 ⊆ 1o)
5 df-2o 6322 . 2 2o = suc 1o
64, 5eleq2s 2235 1 (𝐴 ∈ 2o𝐴 ⊆ 1o)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1481  wss 3076  Tr wtr 4034  suc csuc 4295  1oc1o 6314  2oc2o 6315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298  df-suc 4301  df-1o 6321  df-2o 6322
This theorem is referenced by: (None)
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