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Mirrors > Home > MPE Home > Th. List > clel5OLD | Structured version Visualization version GIF version |
Description: Obsolete version of clel5 3657 as of 19-May-2023. 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.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
clel5OLD | ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ 𝐴) | |
2 | eqeq2 2833 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋)) | |
3 | 2 | adantl 484 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋)) |
4 | eqidd 2822 | . . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑋 = 𝑋) | |
5 | 1, 3, 4 | rspcedvd 3626 | . 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
6 | eleq1a 2908 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑋 = 𝑥 → 𝑋 ∈ 𝐴)) | |
7 | 6 | rexlimiv 3280 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑋 = 𝑥 → 𝑋 ∈ 𝐴) |
8 | 5, 7 | impbii 211 | 1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 |
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-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2814 df-clel 2893 df-ral 3143 df-rex 3144 |
This theorem is referenced by: (None) |
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