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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 5102 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦)) | |
2 | ssbr 5102 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑧 → 𝑥𝐵𝑧)) | |
3 | 1, 2 | anim12d 610 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → (𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧))) |
4 | 3 | eximdv 1914 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧))) |
5 | 4 | ssopab2dv 5430 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} ⊆ {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)}) |
6 | df-coss 35653 | . 2 ⊢ ≀ 𝐴 = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} | |
7 | df-coss 35653 | . 2 ⊢ ≀ 𝐵 = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)} | |
8 | 5, 6, 7 | 3sstr4g 4011 | 1 ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wex 1776 ⊆ wss 3935 class class class wbr 5058 {copab 5120 ≀ ccoss 35447 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3942 df-ss 3951 df-br 5059 df-opab 5121 df-coss 35653 |
This theorem is referenced by: funALTVss 35926 |
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