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Mirrors > Home > MPE Home > Th. List > elabd | Structured version Visualization version GIF version |
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 3665 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
6 | 1, 5 | mpbird 259 | 1 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 {cab 2799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 |
This theorem is referenced by: wfrlem15 7963 lubval 17588 glbval 17601 sursubmefmnd 18055 injsubmefmnd 18056 branmfn 29876 orvcval 31710 nosupfv 33201 rngunsnply 39766 hoidmvlelem1 42871 |
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