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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 33318. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nuliota | ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4934 | . . . 4 ⊢ ∅ ∈ V | |
2 | 1 | eueq1 3512 | . . . . 5 ⊢ ∃!𝑥 𝑥 = ∅ |
3 | eq0 4064 | . . . . . 6 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
4 | 3 | eubii 2621 | . . . . 5 ⊢ (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
5 | 2, 4 | mpbi 220 | . . . 4 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
6 | eleq2 2820 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅)) | |
7 | 6 | notbid 307 | . . . . . 6 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅)) |
8 | 7 | albidv 1990 | . . . . 5 ⊢ (𝑥 = ∅ → (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅)) |
9 | 8 | iota2 6030 | . . . 4 ⊢ ((∅ ∈ V ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) → (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅)) |
10 | 1, 5, 9 | mp2an 710 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∅ ↔ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅) |
11 | noel 4054 | . . 3 ⊢ ¬ 𝑦 ∈ ∅ | |
12 | 10, 11 | mpgbi 1866 | . 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅ |
13 | 12 | eqcomi 2761 | 1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∀wal 1622 = wceq 1624 ∈ wcel 2131 ∃!weu 2599 Vcvv 3332 ∅c0 4050 ℩cio 6002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-nul 4933 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ral 3047 df-rex 3048 df-v 3334 df-sbc 3569 df-dif 3710 df-un 3712 df-nul 4051 df-sn 4314 df-pr 4316 df-uni 4581 df-iota 6004 |
This theorem is referenced by: (None) |
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