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Mirrors > Home > MPE Home > Th. List > clel5OLD | Structured version Visualization version GIF version |
Description: Obsolete version of clel5 3654 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 2832 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋)) | |
3 | 2 | adantl 484 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋)) |
4 | eqidd 2821 | . . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑋 = 𝑋) | |
5 | 1, 3, 4 | rspcedvd 3623 | . 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
6 | eleq1a 2907 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑋 = 𝑥 → 𝑋 ∈ 𝐴)) | |
7 | 6 | rexlimiv 3279 | . 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 1536 ∈ wcel 2113 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-cleq 2813 df-clel 2892 df-ral 3142 df-rex 3143 |
This theorem is referenced by: (None) |
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