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Mirrors > Home > ILE Home > Th. List > clel5 | 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.) (Revised by Steven Nguyen, 19-May-2023.) |
Ref | Expression |
---|---|
clel5 | ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 2485 | . 2 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑋) | |
2 | eqcom 2159 | . . 3 ⊢ (𝑥 = 𝑋 ↔ 𝑋 = 𝑥) | |
3 | 2 | rexbii 2464 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = 𝑋 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
4 | 1, 3 | bitri 183 | 1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1335 ∈ wcel 2128 ∃wrex 2436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-17 1506 ax-ial 1514 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-cleq 2150 df-clel 2153 df-rex 2441 |
This theorem is referenced by: phisum 12103 |
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