Theorem bj-0nelsngl 32152

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 7447). (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 3176	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4252	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2839	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1722	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 32149	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 2985	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 206	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 187	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 2886	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ∉ wnel 2781  ∃wrex 2897  ∅c0 3874  {csn 4125  sngl bj-csngl 32146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-bj-sngl 32147

This theorem is referenced by: (None)