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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nuliota | Structured version Visualization version GIF version | ||
| Description: Definition of the empty set using the definite description binder. See also bj-nuliotaALT 37059. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-nuliota | ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 2 | 1 | eueqi 3715 | . . . . 5 ⊢ ∃!𝑥 𝑥 = ∅ |
| 3 | eq0 4350 | . . . . . 6 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
| 4 | 3 | eubii 2585 | . . . . 5 ⊢ (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
| 5 | 2, 4 | mpbi 230 | . . . 4 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
| 6 | eleq2 2830 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅)) | |
| 7 | 6 | notbid 318 | . . . . . 6 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅)) |
| 8 | 7 | albidv 1920 | . . . . 5 ⊢ (𝑥 = ∅ → (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅)) |
| 9 | 8 | iota2 6550 | . . . 4 ⊢ ((∅ ∈ V ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅)) |
| 10 | 1, 5, 9 | mp2an 692 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅) |
| 11 | noel 4338 | . . 3 ⊢ ¬ 𝑦 ∈ ∅ | |
| 12 | 10, 11 | mpgbi 1798 | . 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅ |
| 13 | 12 | eqcomi 2746 | 1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃!weu 2568 Vcvv 3480 ∅c0 4333 ℩cio 6512 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 |
| This theorem is referenced by: (None) |
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