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Mirrors > Home > MPE Home > Th. List > eq0f | Structured version Visualization version GIF version |
Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.) |
Ref | Expression |
---|---|
eq0f.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
eq0f | ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑥∅ | |
3 | 1, 2 | cleqf 2932 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
4 | noel 4344 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
5 | 4 | nbn 372 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
6 | 5 | albii 1816 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
7 | 3, 6 | bitr4i 278 | 1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1535 = wceq 1537 ∈ wcel 2106 Ⅎwnfc 2888 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-dif 3966 df-nul 4340 |
This theorem is referenced by: neq0f 4354 ab0ALT 4387 bnj1476 34840 stoweidlem34 45990 |
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