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Theorem ss1o0el1o 6911
Description: Reformulation of ss1o0el1 4197 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
StepHypRef Expression
1 df1o2 6429 . . . 4  |-  1o  =  { (/) }
21eqcomi 2181 . . 3  |-  { (/) }  =  1o
32sseq2i 3182 . 2  |-  ( A 
C_  { (/) }  <->  A  C_  1o )
4 ss1o0el1 4197 . . 3  |-  ( A 
C_  { (/) }  ->  (
(/)  e.  A  <->  A  =  { (/) } ) )
52eqeq2i 2188 . . 3  |-  ( A  =  { (/) }  <->  A  =  1o )
64, 5bitrdi 196 . 2  |-  ( A 
C_  { (/) }  ->  (
(/)  e.  A  <->  A  =  1o ) )
73, 6sylbir 135 1  |-  ( A 
C_  1o  ->  ( (/)  e.  A  <->  A  =  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148    C_ wss 3129   (/)c0 3422   {csn 3592   1oc1o 6409
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 4129
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 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-suc 4371  df-1o 6416
This theorem is referenced by:  pw1dc1  6912
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