![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > el | Structured version Visualization version GIF version |
Description: Any set is an element of some other set. See elALT 5446 for a shorter proof using more axioms, and see elALT2 5373 for a proof that uses ax-9 2109 and ax-pow 5369 instead of ax-pr 5433. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5433 instead of ax-9 2109 and ax-pow 5369. (Revised by BTernaryTau, 2-Dec-2024.) |
Ref | Expression |
---|---|
el | ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pr 5433 | . 2 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) | |
2 | pm4.25 903 | . . . . . 6 ⊢ (𝑧 = 𝑥 ↔ (𝑧 = 𝑥 ∨ 𝑧 = 𝑥)) | |
3 | 2 | imbi1i 348 | . . . . 5 ⊢ ((𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ ((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦)) |
4 | 3 | albii 1814 | . . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦)) |
5 | elequ1 2106 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) | |
6 | 5 | equsalvw 2000 | . . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦) |
7 | 4, 6 | bitr3i 276 | . . 3 ⊢ (∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦) |
8 | 7 | exbii 1843 | . 2 ⊢ (∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦 𝑥 ∈ 𝑦) |
9 | 1, 8 | mpbi 229 | 1 ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 ∀wal 1532 ∃wex 1774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1775 |
This theorem is referenced by: sels 5444 dtruOLD 5447 dmep 5930 axpownd 10644 zfcndinf 10661 distel 35627 |
Copyright terms: Public domain | W3C validator |