Theorem bj-0nelsngl 36966

Description: The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 8437). (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-0nelsngl	⊢ ∅ ∉ sngl 𝐴

Proof of Theorem bj-0nelsngl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4743	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2983	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1800	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 36963	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3060	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 217	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 197	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3033	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ∉ wnel 3030  ∃wrex 3054  ∅c0 4299  {csn 4592  sngl bj-csngl 36960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-nel 3031  df-rex 3055  df-v 3452  df-dif 3920  df-un 3922  df-nul 4300  df-sn 4593  df-pr 4595  df-bj-sngl 36961

This theorem is referenced by: (None)