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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 5038 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 4975 | . 2 ⊢ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
2 | ax9 2158 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥)) | |
3 | 2 | alrimiv 2007 | . . . 4 ⊢ (𝑧 = 𝑥 → ∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥)) |
4 | ax8 2151 | . . . 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 | spimv 2419 | . 2 ⊢ (∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) |
7 | 1, 6 | eximii 1912 | 1 ⊢ ∃𝑦 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1629 ∃wex 1852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-12 2203 ax-13 2408 ax-pow 4974 |
This theorem depends on definitions: df-bi 197 df-an 383 df-ex 1853 df-nf 1858 |
This theorem is referenced by: dtru 4988 dvdemo2 5031 axpownd 9625 zfcndinf 9642 domep 32034 distel 32045 |
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