Theorem bj-0nelsngl 35088

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 8267). (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 3426	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4709	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3000	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1804	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 35085	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3167	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 216	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 196	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3051	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ∉ wnel 3048  ∃wrex 3064  ∅c0 4253  {csn 4558  sngl bj-csngl 35082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-nel 3049  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-pr 4561  df-bj-sngl 35083

This theorem is referenced by: (None)