Theorem ss1o0el1o 6914

Description: Reformulation of ss1o0el1 4199 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 6432	. . . 4 |- 1o = { (/) }
2	1	eqcomi 2181	. . 3 |- { (/) } = 1o
3	2	sseq2i 3184	. 2 |- ( A C_ { (/) } <-> A C_ 1o )
4		ss1o0el1 4199	. . 3 |- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )
5	2	eqeq2i 2188	. . 3 |- ( A = { (/) } <-> A = 1o )
6	4, 5	bitrdi 196	. 2 |- ( A C_ { (/) } -> ( (/) e. A <-> A = 1o ) )
7	3, 6	sylbir 135	1 |- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   C_ wss 3131   (/)c0 3424   {csn 3594   1oc1o 6412

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-ext 2159  ax-nul 4131

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-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-suc 4373  df-1o 6419

This theorem is referenced by:  pw1dc1  6915