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Mirrors > Home > ILE Home > Th. List > el | GIF version |
Description: Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
el | ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfpow 4161 | . 2 ⊢ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
2 | ax-14 2144 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥)) | |
3 | 2 | alrimiv 1867 | . . . 4 ⊢ (𝑧 = 𝑥 → ∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥)) |
4 | ax-13 2143 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦)) | |
5 | 3, 4 | embantd 56 | . . 3 ⊢ (𝑧 = 𝑥 → ((∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)) |
6 | 5 | spimv 1804 | . 2 ⊢ (∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) |
7 | 1, 6 | eximii 1595 | 1 ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 ∃wex 1485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-13 2143 ax-14 2144 ax-pow 4160 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: dtruarb 4177 |
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