Theorem bj-disjsn01 33132

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 33131 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 7663	. . 3 ⊢ 1𝑜 ≠ ∅
2	1	necomi 2918	. 2 ⊢ ∅ ≠ 1𝑜
3		disjsn2 4322	. 2 ⊢ (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1𝑜}) = ∅

Syntax hints:   = wceq 1564   ≠ wne 2864   ∩ cin 3647  ∅c0 3991  {csn 4253  1_𝑜c1o 7641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-nul 4865

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-v 3274  df-dif 3651  df-un 3653  df-in 3655  df-nul 3992  df-sn 4254  df-suc 5810  df-1o 7648

This theorem is referenced by:  bj-2upln1upl  33207