Theorem bj-el2oss1o 14929

Description: Shorter proof of el2oss1o 6462 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 6442	. . . 4 |- 1o e. On
2	1	ontrci 4442	. . 3 |- Tr 1o
3		trsucss 4438	. . 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 6436	. 2 |- 2o = suc 1o
6	4, 5	eleq2s 2284	1 |- ( A e. 2o -> A C_ 1o )

Syntax hints:   -> wi 4   e. wcel 2160   C_ wss 3144   Tr wtr 4116   suc csuc 4380   1oc1o 6428   2oc2o 6429

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386  df-1o 6435  df-2o 6436

This theorem is referenced by: (None)