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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 8094). (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-0nelsngl | ⊢ ∅ ∉ sngl 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3496 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snnz 4703 | . . . . 5 ⊢ {𝑥} ≠ ∅ |
3 | 2 | nesymi 3071 | . . . 4 ⊢ ¬ ∅ = {𝑥} |
4 | 3 | nex 1795 | . . 3 ⊢ ¬ ∃𝑥∅ = {𝑥} |
5 | bj-elsngl 34273 | . . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥}) | |
6 | rexex 3238 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥}) | |
7 | 5, 6 | sylbi 219 | . . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥}) |
8 | 4, 7 | mto 199 | . 2 ⊢ ¬ ∅ ∈ sngl 𝐴 |
9 | 8 | nelir 3124 | 1 ⊢ ∅ ∉ sngl 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 ∃wex 1774 ∈ wcel 2108 ∉ wnel 3121 ∃wrex 3137 ∅c0 4289 {csn 4559 sngl bj-csngl 34270 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-rex 3142 df-v 3495 df-dif 3937 df-un 3939 df-nul 4290 df-sn 4560 df-pr 4562 df-bj-sngl 34271 |
This theorem is referenced by: (None) |
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