Theorem ss1o0el1o 6878

Description: Reformulation of ss1o0el1 4176 using 1o instead of { (/) }. (Contributed by BJ, 9-Aug-2024.)

Assertion

Ref	Expression
ss1o0el1o	|- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )

Proof of Theorem ss1o0el1o

Step	Hyp	Ref	Expression
1		df1o2 6397	. . . 4 |- 1o = { (/) }
2	1	eqcomi 2169	. . 3 |- { (/) } = 1o
3	2	sseq2i 3169	. 2 |- ( A C_ { (/) } <-> A C_ 1o )
4		ss1o0el1 4176	. . 3 |- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )
5	2	eqeq2i 2176	. . 3 |- ( A = { (/) } <-> A = 1o )
6	4, 5	bitrdi 195	. 2 |- ( A C_ { (/) } -> ( (/) e. A <-> A = 1o ) )
7	3, 6	sylbir 134	1 |- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   C_ wss 3116   (/)c0 3409   {csn 3576   1oc1o 6377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-suc 4349  df-1o 6384

This theorem is referenced by:  pw1dc1  6879