Theorem bj-el2oss1o 14386

Description: Shorter proof of el2oss1o 6441 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

Step	Hyp	Ref	Expression
1		1on 6421	. . . 4 |- 1o e. On
2	1	ontrci 4426	. . 3 |- Tr 1o
3		trsucss 4422	. . 3 |- ( Tr 1o -> ( A e. suc 1o -> A C_ 1o ) )
4	2, 3	ax-mp 5	. 2 |- ( A e. suc 1o -> A C_ 1o )
5		df-2o 6415	. 2 |- 2o = suc 1o
6	4, 5	eleq2s 2272	1 |- ( A e. 2o -> A C_ 1o )

Syntax hints:   -> wi 4   e. wcel 2148   C_ wss 3129   Tr wtr 4100   suc csuc 4364   1oc1o 6407   2oc2o 6408

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 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432

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 4101  df-iord 4365  df-on 4367  df-suc 4370  df-1o 6414  df-2o 6415

This theorem is referenced by: (None)