Theorem bj-disjsn01 34266

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 34265 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 8121	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3072	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4650	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1537   ≠ wne 3018   ∩ cin 3937  ∅c0 4293  {csn 4569  1_oc1o 8097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-nul 4294  df-sn 4570  df-suc 6199  df-1o 8104

This theorem is referenced by: (None)