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Mirrors > Home > MPE Home > Th. List > axexte | Structured version Visualization version GIF version |
Description: The axiom of extensionality (ax-ext 2793) restated so that it postulates the existence of a set 𝑧 given two arbitrary sets 𝑥 and 𝑦. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.) |
Ref | Expression |
---|---|
axexte | ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-ext 2793 | . 2 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | |
2 | 19.36v 1994 | . 2 ⊢ (∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) ↔ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)) | |
3 | 1, 2 | mpbir 233 | 1 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-ex 1781 |
This theorem is referenced by: (None) |
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