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Mirrors > Home > MPE Home > Th. List > clel5 | Structured version Visualization version GIF version |
Description: Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.) |
Ref | Expression |
---|---|
clel5 | ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ 𝐴) | |
2 | eqeq2 2772 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋)) | |
3 | 2 | adantl 473 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋)) |
4 | eqidd 2762 | . . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑋 = 𝑋) | |
5 | 1, 3, 4 | rspcedvd 3457 | . 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
6 | eleq1a 2835 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑋 = 𝑥 → 𝑋 ∈ 𝐴)) | |
7 | 6 | rexlimiv 3166 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑋 = 𝑥 → 𝑋 ∈ 𝐴) |
8 | 5, 7 | impbii 199 | 1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1632 ∈ wcel 2140 ∃wrex 3052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ral 3056 df-rex 3057 df-v 3343 |
This theorem is referenced by: dfss5 4008 disjunsn 29736 |
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