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Mirrors > Home > MPE Home > Th. List > Mathboxes > clss2lem | Structured version Visualization version GIF version |
Description: The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.) |
Ref | Expression |
---|---|
clss2lem.1 | ⊢ (𝜑 → (𝜒 → 𝜓)) |
Ref | Expression |
---|---|
clss2lem | ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clss2lem.1 | . . . . 5 ⊢ (𝜑 → (𝜒 → 𝜓)) | |
2 | 1 | adantld 491 | . . . 4 ⊢ (𝜑 → ((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓)) |
3 | 2 | alrimiv 1905 | . . 3 ⊢ (𝜑 → ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓)) |
4 | pm5.3 573 | . . . . 5 ⊢ (((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ ((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓))) | |
5 | 4 | albii 1801 | . . . 4 ⊢ (∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓))) |
6 | ss2ab 3960 | . . . 4 ⊢ ({𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ↔ ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓))) | |
7 | 5, 6 | bitr4i 279 | . . 3 ⊢ (∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) |
8 | 3, 7 | sylib 219 | . 2 ⊢ (𝜑 → {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) |
9 | intss 4803 | . 2 ⊢ ({𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)}) | |
10 | 8, 9 | syl 17 | 1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1520 {cab 2775 ⊆ wss 3859 ∩ cint 4782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-in 3866 df-ss 3874 df-int 4783 |
This theorem is referenced by: (None) |
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