Theorem bj-el2oss1o 16716

Description: Shorter proof of el2oss1o 6706 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 6684	. . . 4 |- 1o e. On
2	1	ontrci 4567	. . 3 |- Tr 1o
3		trsucss 4563	. . 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 6678	. 2 |- 2o = suc 1o
6	4, 5	eleq2s 2333	1 |- ( A e. 2o -> A C_ 1o )

Syntax hints:   -> wi 4   e. wcel 2209   C_ wss 3220   Tr wtr 4224   suc csuc 4505   1oc1o 6670   2oc2o 6671

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-1o 6677  df-2o 6678

This theorem is referenced by: (None)