Theorem bj-0nelsngl 32959

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 7560). (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 3203	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4309	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2851	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1731	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 32956	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3002	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 207	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 188	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 2900	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ∉ wnel 2897  ∃wrex 2913  ∅c0 3915  {csn 4177  sngl bj-csngl 32953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-bj-sngl 32954

This theorem is referenced by: (None)