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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 3663 | . . 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 205 = wceq 1534 ∈ wcel 2099 {cab 2704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 |
This theorem is referenced by: intidg 5453 wfrlem15OLD 8337 lubval 18339 glbval 18352 sursubmefmnd 18839 injsubmefmnd 18840 nosupfv 27626 branmfn 31902 orvcval 34013 sticksstones3 41552 rngunsnply 42519 hoidmvlelem1 45906 cfsetsnfsetf 46363 |
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