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Mirrors > Home > MPE Home > Th. List > Mathboxes > currysetlem2 | Structured version Visualization version GIF version |
Description: Lemma for currysetALT 36933. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
currysetlem2.def | ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} |
Ref | Expression |
---|---|
currysetlem2 | ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | currysetlem2.def | . . . 4 ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} | |
2 | 1 | currysetlem1 36930 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
3 | 2 | biimpd 229 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → (𝑋 ∈ 𝑋 → 𝜑))) |
4 | 3 | pm2.43d 53 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-v 3480 |
This theorem is referenced by: currysetlem3 36932 |
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