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Theorem bj-el2oss1o 14408
Description: Shorter proof of el2oss1o 6443 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 6423 . . . 4 1o ∈ On
21ontrci 4427 . . 3 Tr 1o
3 trsucss 4423 . . 3 (Tr 1o → (𝐴 ∈ suc 1o𝐴 ⊆ 1o))
42, 3ax-mp 5 . 2 (𝐴 ∈ suc 1o𝐴 ⊆ 1o)
5 df-2o 6417 . 2 2o = suc 1o
64, 5eleq2s 2272 1 (𝐴 ∈ 2o𝐴 ⊆ 1o)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  wss 3129  Tr wtr 4101  suc csuc 4365  1oc1o 6409  2oc2o 6410
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371  df-1o 6416  df-2o 6417
This theorem is referenced by: (None)
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