Theorem bj-disjsn01 37449

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9560 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-disjsn01	⊢ ({∅} ∩ {1o}) = ∅

Proof of Theorem bj-disjsn01

Step	Hyp	Ref	Expression
1		1n0 8460	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3014	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4674	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1563   ≠ wne 2960   ∩ cin 3906  ∅c0 4288  {csn 4585  1_oc1o 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-nul 4289  df-sn 4586  df-suc 6356  df-1o 8441

This theorem is referenced by: (None)