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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.) Remove use of ax-10 2129, ax-11 2146, and ax-12 2166. (Revised by Steven Nguyen, 19-May-2023.) |
Ref | Expression |
---|---|
clel5 | ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 3220 | . 2 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑋) | |
2 | eqcom 2732 | . . 3 ⊢ (𝑥 = 𝑋 ↔ 𝑋 = 𝑥) | |
3 | 2 | rexbii 3083 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = 𝑋 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
4 | 1, 3 | bitri 274 | 1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-cleq 2717 df-clel 2802 df-rex 3060 |
This theorem is referenced by: dfss5 4263 iunid 5064 4fvwrd4 13654 wrdlen1 14538 phisum 16760 symgmov1 19350 disjunsn 32457 rp-abid 42919 |
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