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Theorem bj-0nelsngl 32633
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 7512). (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.
StepHypRef Expression
1 vex 3192 . . . . . 6 𝑥 ∈ V
21snnz 4284 . . . . 5 {𝑥} ≠ ∅
32nesymi 2847 . . . 4 ¬ ∅ = {𝑥}
43nex 1728 . . 3 ¬ ∃𝑥∅ = {𝑥}
5 bj-elsngl 32630 . . . 4 (∅ ∈ sngl 𝐴 ↔ ∃𝑥𝐴 ∅ = {𝑥})
6 rexex 2997 . . . 4 (∃𝑥𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
75, 6sylbi 207 . . 3 (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
84, 7mto 188 . 2 ¬ ∅ ∈ sngl 𝐴
98nelir 2896 1 ∅ ∉ sngl 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wex 1701  wcel 1987  wnel 2893  wrex 2908  c0 3896  {csn 4153  sngl bj-csngl 32627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-v 3191  df-dif 3562  df-un 3564  df-nul 3897  df-sn 4154  df-pr 4156  df-bj-sngl 32628
This theorem is referenced by: (None)
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