Theorem bj-disjsn01 32937

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 32936 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-disjsn01	|- ( { (/) } i^i { 1o } ) = (/)

Proof of Theorem bj-disjsn01

Step	Hyp	Ref	Expression
1		1n0 7575	. . 3 |- 1o =/= (/)
2	1	necomi 2848	. 2 |- (/) =/= 1o
3		disjsn2 4247	. 2 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
4	2, 3	ax-mp 5	1 |- ( { (/) } i^i { 1o } ) = (/)

Syntax hints:   = wceq 1483   =/= wne 2794   i^i cin 3573   (/)c0 3915   {csn 4177   1oc1o 7553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-suc 5729  df-1o 7560

This theorem is referenced by:  bj-2upln1upl  33012