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Mirrors > Home > MPE Home > Th. List > el | Structured version Visualization version GIF version |
Description: Every set is an element of some other set. See elALT 5191 for a shorter proof using more axioms. (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 5120 | . 2 ⊢ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
2 | ax9 2063 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥)) | |
3 | 2 | alrimiv 1886 | . . . 4 ⊢ (𝑧 = 𝑥 → ∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥)) |
4 | ax8 2056 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦)) | |
5 | 3, 4 | embantd 59 | . . 3 ⊢ (𝑧 = 𝑥 → ((∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)) |
6 | 5 | spimvw 1954 | . 2 ⊢ (∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) |
7 | 1, 6 | eximii 1799 | 1 ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1505 ∃wex 1742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-pow 5119 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 |
This theorem is referenced by: dtru 5124 dvdemo2 5128 axpownd 9821 zfcndinf 9838 domep 32555 distel 32566 |
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