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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-0nelsngl | Structured version Visualization version GIF version |
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 8486). (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-0nelsngl | ⊢ ∅ ∉ sngl 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3475 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snnz 4781 | . . . . 5 ⊢ {𝑥} ≠ ∅ |
3 | 2 | nesymi 2995 | . . . 4 ⊢ ¬ ∅ = {𝑥} |
4 | 3 | nex 1795 | . . 3 ⊢ ¬ ∃𝑥∅ = {𝑥} |
5 | bj-elsngl 36447 | . . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥}) | |
6 | rexex 3073 | . . . 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 3046 | 1 ⊢ ∅ ∉ sngl 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∉ wnel 3043 ∃wrex 3067 ∅c0 4323 {csn 4629 sngl bj-csngl 36444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-nel 3044 df-rex 3068 df-v 3473 df-dif 3950 df-un 3952 df-nul 4324 df-sn 4630 df-pr 4632 df-bj-sngl 36445 |
This theorem is referenced by: (None) |
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