|   | Mathbox for BJ | < Previous  
      Next > Nearby theorems | |
| 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 8506). (Contributed by BJ, 6-Oct-2018.) | 
| Ref | Expression | 
|---|---|
| bj-0nelsngl | ⊢ ∅ ∉ sngl 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 2 | 1 | snnz 4776 | . . . . 5 ⊢ {𝑥} ≠ ∅ | 
| 3 | 2 | nesymi 2998 | . . . 4 ⊢ ¬ ∅ = {𝑥} | 
| 4 | 3 | nex 1800 | . . 3 ⊢ ¬ ∃𝑥∅ = {𝑥} | 
| 5 | bj-elsngl 36969 | . . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥}) | |
| 6 | rexex 3076 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥}) | |
| 7 | 5, 6 | sylbi 217 | . . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥}) | 
| 8 | 4, 7 | mto 197 | . 2 ⊢ ¬ ∅ ∈ sngl 𝐴 | 
| 9 | 8 | nelir 3049 | 1 ⊢ ∅ ∉ sngl 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∉ wnel 3046 ∃wrex 3070 ∅c0 4333 {csn 4626 sngl bj-csngl 36966 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 df-bj-sngl 36967 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |