Theorem bj-el2oss1o 16138

Description: Shorter proof of el2oss1o 6589 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 6569	. . . 4 |- 1o e. On
2	1	ontrci 4518	. . 3 |- Tr 1o
3		trsucss 4514	. . 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 6563	. 2 |- 2o = suc 1o
6	4, 5	eleq2s 2324	1 |- ( A e. 2o -> A C_ 1o )

Syntax hints:   -> wi 4   e. wcel 2200   C_ wss 3197   Tr wtr 4182   suc csuc 4456   1oc1o 6555   2oc2o 6556

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462  df-1o 6562  df-2o 6563

This theorem is referenced by: (None)