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Mirrors > Home > MPE Home > Th. List > ssdif | Structured version Visualization version GIF version |
Description: Difference law for subsets. (Contributed by NM, 28-May-1998.) |
Ref | Expression |
---|---|
ssdif | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3597 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | anim1d 588 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))) |
3 | eldif 3584 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)) | |
4 | eldif 3584 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) | |
5 | 2, 3, 4 | 3imtr4g 285 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐶) → 𝑥 ∈ (𝐵 ∖ 𝐶))) |
6 | 5 | ssrdv 3609 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∈ wcel 1990 ∖ cdif 3571 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 |
This theorem is referenced by: ssdifd 3746 php 8144 pssnn 8178 fin1a2lem13 9234 axcclem 9279 isercolllem3 14397 mvdco 17865 dprdres 18427 dpjidcl 18457 ablfac1eulem 18471 lspsnat 19145 lbsextlem2 19159 lbsextlem3 19160 mplmonmul 19464 cnsubdrglem 19797 clsconn 21233 2ndcdisj2 21260 kqdisj 21535 nulmbl2 23304 i1f1 23457 itg11 23458 itg1climres 23481 limcresi 23649 dvreslem 23673 dvres2lem 23674 dvaddbr 23701 dvmulbr 23702 lhop 23779 elqaa 24077 difres 29413 imadifxp 29414 xrge00 29686 eulerpartlemmf 30437 eulerpartlemgf 30441 bj-2upln1upl 33012 mblfinlem3 33448 mblfinlem4 33449 ismblfin 33450 cnambfre 33458 divrngidl 33827 cntzsdrg 37772 radcnvrat 38513 fourierdlem62 40385 |
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