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Mirrors > Home > MPE Home > Th. List > Mathboxes > clublem | Structured version Visualization version GIF version |
Description: If a superset 𝑌 of 𝑋 possesses the property parameterized in 𝑥 in 𝜓, then 𝑌 is a superset of the closure of that property for the set 𝑋. (Contributed by RP, 23-Jul-2020.) |
Ref | Expression |
---|---|
clublem.y | ⊢ (𝜑 → 𝑌 ∈ V) |
clublem.sub | ⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜒)) |
clublem.sup | ⊢ (𝜑 → 𝑋 ⊆ 𝑌) |
clublem.maj | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
clublem | ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clublem.sup | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑌) | |
2 | clublem.maj | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | clublem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ V) | |
4 | 3 | a1d 25 | . . . 4 ⊢ (𝜑 → ((𝑋 ⊆ 𝑌 ∧ 𝜒) → 𝑌 ∈ V)) |
5 | clublem.sub | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜒)) | |
6 | 5 | cleq2lem 40892 | . . . . 5 ⊢ (𝑥 = 𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ 𝑌 ∧ 𝜒))) |
7 | 6 | elab3g 3594 | . . . 4 ⊢ (((𝑋 ⊆ 𝑌 ∧ 𝜒) → 𝑌 ∈ V) → (𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ↔ (𝑋 ⊆ 𝑌 ∧ 𝜒))) |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ↔ (𝑋 ⊆ 𝑌 ∧ 𝜒))) |
9 | 1, 2, 8 | mpbir2and 713 | . 2 ⊢ (𝜑 → 𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) |
10 | intss1 4874 | . 2 ⊢ (𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌) | |
11 | 9, 10 | syl 17 | 1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {cab 2714 Vcvv 3408 ⊆ wss 3866 ∩ cint 4859 |
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 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-in 3873 df-ss 3883 df-int 4860 |
This theorem is referenced by: mptrcllem 40897 trclubgNEW 40902 |
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