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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cossss | Structured version Visualization version GIF version | ||
| Description: Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.) |
| Ref | Expression |
|---|---|
| cossss | ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssbr 5187 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦)) | |
| 2 | ssbr 5187 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑧 → 𝑥𝐵𝑧)) | |
| 3 | 1, 2 | anim12d 609 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → (𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧))) |
| 4 | 3 | eximdv 1917 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧))) |
| 5 | 4 | ssopab2dv 5556 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} ⊆ {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)}) |
| 6 | df-coss 38412 | . 2 ⊢ ≀ 𝐴 = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} | |
| 7 | df-coss 38412 | . 2 ⊢ ≀ 𝐵 = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)} | |
| 8 | 5, 6, 7 | 3sstr4g 4037 | 1 ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ⊆ wss 3951 class class class wbr 5143 {copab 5205 ≀ ccoss 38182 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-br 5144 df-opab 5206 df-coss 38412 |
| This theorem is referenced by: funALTVss 38700 |
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