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Mirrors > Home > MPE Home > Th. List > elrabiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of elrabi 3609 as of 5-Aug-2024. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
elrabiOLD | ⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelab 2881 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑))) | |
2 | eleq1 2826 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉)) | |
3 | 2 | anbi1d 633 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ (𝐴 ∈ 𝑉 ∧ 𝜑))) |
4 | 3 | simprbda 502 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉) |
5 | 4 | exlimiv 1938 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉) |
6 | 1, 5 | sylbi 220 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} → 𝐴 ∈ 𝑉) |
7 | df-rab 3071 | . 2 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} | |
8 | 6, 7 | eleq2s 2857 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2111 {cab 2715 {crab 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2176 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 |
This theorem is referenced by: (None) |
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