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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 3614 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
6 | 1, 5 | mpbird 260 | 1 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 |
This theorem is referenced by: wfrlem15 7952 lubval 17586 glbval 17599 sursubmefmnd 18053 injsubmefmnd 18054 branmfn 29888 orvcval 31825 nosupfv 33319 rngunsnply 40117 hoidmvlelem1 43234 |
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