Theorem ss1o0el1o 6854

Description: Reformulation of ss1o0el1 4158 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 6373	. . . 4 |- 1o = { (/) }
2	1	eqcomi 2161	. . 3 |- { (/) } = 1o
3	2	sseq2i 3155	. 2 |- ( A C_ { (/) } <-> A C_ 1o )
4		ss1o0el1 4158	. . 3 |- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )
5	2	eqeq2i 2168	. . 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 1335   e. wcel 2128   C_ wss 3102   (/)c0 3394   {csn 3560   1oc1o 6353

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4090

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-suc 4331  df-1o 6360

This theorem is referenced by:  pw1dc1  6855