Theorem bj-0nelsngl 36972

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 8506). (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 3484	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4776	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2998	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1800	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 36969	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3076	. . . 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 3049	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ∉ wnel 3046  ∃wrex 3070  ∅c0 4333  {csn 4626  sngl bj-csngl 36966

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 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629  df-bj-sngl 36967

This theorem is referenced by: (None)