Theorem bj-0nelsngl 37174

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 8399). (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 3445	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4734	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2990	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1802	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 37171	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3067	. . . 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 3040	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ∉ wnel 3037  ∃wrex 3061  ∅c0 4286  {csn 4581  sngl bj-csngl 37168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-nel 3038  df-rex 3062  df-v 3443  df-dif 3905  df-un 3907  df-nul 4287  df-sn 4582  df-pr 4584  df-bj-sngl 37169

This theorem is referenced by: (None)