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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 33970. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nuliota | ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5109 | . . . 4 ⊢ ∅ ∈ V | |
2 | 1 | eueqi 3641 | . . . . 5 ⊢ ∃!𝑥 𝑥 = ∅ |
3 | eq0 4234 | . . . . . 6 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
4 | 3 | eubii 2632 | . . . . 5 ⊢ (∃!𝑥 𝑥 = ∅ ↔ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
5 | 2, 4 | mpbi 231 | . . . 4 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
6 | eleq2 2873 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅)) | |
7 | 6 | notbid 319 | . . . . . 6 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅)) |
8 | 7 | albidv 1902 | . . . . 5 ⊢ (𝑥 = ∅ → (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ ∅)) |
9 | 8 | iota2 6222 | . . . 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 4222 | . . 3 ⊢ ¬ 𝑦 ∈ ∅ | |
12 | 10, 11 | mpgbi 1784 | . 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) = ∅ |
13 | 12 | eqcomi 2806 | 1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∀wal 1523 = wceq 1525 ∈ wcel 2083 ∃!weu 2613 Vcvv 3440 ∅c0 4217 ℩cio 6194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-nul 5108 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-nul 4218 df-sn 4479 df-pr 4481 df-uni 4752 df-iota 6196 |
This theorem is referenced by: (None) |
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