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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 7826). (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-0nelsngl | ⊢ ∅ ∉ sngl 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3417 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | snnz 4528 | . . . . 5 ⊢ {𝑥} ≠ ∅ |
3 | 2 | nesymi 3056 | . . . 4 ⊢ ¬ ∅ = {𝑥} |
4 | 3 | nex 1901 | . . 3 ⊢ ¬ ∃𝑥∅ = {𝑥} |
5 | bj-elsngl 33478 | . . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥}) | |
6 | rexex 3210 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥}) | |
7 | 5, 6 | sylbi 209 | . . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥}) |
8 | 4, 7 | mto 189 | . 2 ⊢ ¬ ∅ ∈ sngl 𝐴 |
9 | 8 | nelir 3105 | 1 ⊢ ∅ ∉ sngl 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∃wex 1880 ∈ wcel 2166 ∉ wnel 3102 ∃wrex 3118 ∅c0 4144 {csn 4397 sngl bj-csngl 33475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-v 3416 df-dif 3801 df-un 3803 df-nul 4145 df-sn 4398 df-pr 4400 df-bj-sngl 33476 |
This theorem is referenced by: (None) |
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