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| Description: Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝐴. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 23-Mar-2024.) | 
| Ref | Expression | 
|---|---|
| elabd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| elabd.2 | ⊢ (𝜑 → 𝜒) | 
| elabd.3 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| elabd | ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elabd.2 | . 2 ⊢ (𝜑 → 𝜒) | |
| 2 | elabd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 3 | elabd.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | |
| 4 | 3 | elabg 3675 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) | 
| 5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) | 
| 6 | 1, 5 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {cab 2713 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: intidg 5461 wfrlem15OLD 8364 lubval 18402 glbval 18415 sursubmefmnd 18910 injsubmefmnd 18911 nosupfv 27752 branmfn 32125 orvcval 34461 r1peuqusdeg1 35649 sticksstones3 42150 rngunsnply 43186 hoidmvlelem1 46615 cfsetsnfsetf 47075 | 
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