Theorem ss1o0el1o 7173

Description: Reformulation of ss1o0el1 4310 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 6661	. . . 4 |- 1o = { (/) }
2	1	eqcomi 2236	. . 3 |- { (/) } = 1o
3	2	sseq2i 3265	. 2 |- ( A C_ { (/) } <-> A C_ 1o )
4		ss1o0el1 4310	. . 3 |- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )
5	2	eqeq2i 2243	. . 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 1398   e. wcel 2203   C_ wss 3211   (/)c0 3508   {csn 3689   1oc1o 6640

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-nul 4236

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-suc 4492  df-1o 6647

This theorem is referenced by:  pw1dc1  7174