Theorem ss1o0el1o 6983

Description: Reformulation of ss1o0el1 4231 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 6496	. . . 4 |- 1o = { (/) }
2	1	eqcomi 2200	. . 3 |- { (/) } = 1o
3	2	sseq2i 3211	. 2 |- ( A C_ { (/) } <-> A C_ 1o )
4		ss1o0el1 4231	. . 3 |- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )
5	2	eqeq2i 2207	. . 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 1364   e. wcel 2167   C_ wss 3157   (/)c0 3451   {csn 3623   1oc1o 6476

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4160

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-suc 4407  df-1o 6483

This theorem is referenced by:  pw1dc1  6984