Theorem bj-disjsn01 36920

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9675 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 8546	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3001	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4737	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1537   ≠ wne 2946   ∩ cin 3975  ∅c0 4352  {csn 4648  1_oc1o 8517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-nul 4353  df-sn 4649  df-suc 6403  df-1o 8524

This theorem is referenced by: (None)