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| 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 5445 for a shorter proof using more axioms, and see elALT2 5369 for a proof that uses ax-9 2118 and ax-pow 5365 instead of ax-pr 5432. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5432 instead of ax-9 2118 and ax-pow 5365. (Revised by BTernaryTau, 2-Dec-2024.) |
| Ref | Expression |
|---|---|
| el | ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-pr 5432 | . 2 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) | |
| 2 | pm4.25 906 | . . . . . 6 ⊢ (𝑧 = 𝑥 ↔ (𝑧 = 𝑥 ∨ 𝑧 = 𝑥)) | |
| 3 | 2 | imbi1i 349 | . . . . 5 ⊢ ((𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ ((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦)) |
| 4 | 3 | albii 1819 | . . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦)) |
| 5 | elequ1 2115 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) | |
| 6 | 5 | equsalvw 2003 | . . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦) |
| 7 | 4, 6 | bitr3i 277 | . . 3 ⊢ (∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦) |
| 8 | 7 | exbii 1848 | . 2 ⊢ (∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦 𝑥 ∈ 𝑦) |
| 9 | 1, 8 | mpbi 230 | 1 ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 ∀wal 1538 ∃wex 1779 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 |
| This theorem is referenced by: sels 5443 dtruOLD 5446 dmep 5934 axpownd 10641 zfcndinf 10658 distel 35804 |
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