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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 36242. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nuliota | ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5306 | . . . 4 ⊢ ∅ ∈ V | |
2 | 1 | eueqi 3704 | . . . . 5 ⊢ ∃!𝑥 𝑥 = ∅ |
3 | eq0 4342 | . . . . . 6 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
4 | 3 | eubii 2577 | . . . . 5 ⊢ (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
5 | 2, 4 | mpbi 229 | . . . 4 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
6 | eleq2 2820 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅)) | |
7 | 6 | notbid 317 | . . . . . 6 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅)) |
8 | 7 | albidv 1921 | . . . . 5 ⊢ (𝑥 = ∅ → (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅)) |
9 | 8 | iota2 6531 | . . . 4 ⊢ ((∅ ∈ V ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅)) |
10 | 1, 5, 9 | mp2an 688 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅) |
11 | noel 4329 | . . 3 ⊢ ¬ 𝑦 ∈ ∅ | |
12 | 10, 11 | mpgbi 1798 | . 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅ |
13 | 12 | eqcomi 2739 | 1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1537 = wceq 1539 ∈ wcel 2104 ∃!weu 2560 Vcvv 3472 ∅c0 4321 ℩cio 6492 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rex 3069 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-uni 4908 df-iota 6494 |
This theorem is referenced by: (None) |
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