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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 258 | 1 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {cab 2796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 |
This theorem is referenced by: wfrlem15 7958 lubval 17582 glbval 17595 branmfn 29809 orvcval 31614 nosupfv 33103 rngunsnply 39651 hoidmvlelem1 42754 sursubmefmnd 43993 injsubmefmnd 43994 |
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