![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > relssi | Structured version Visualization version GIF version |
Description: Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.) |
Ref | Expression |
---|---|
relssi.1 | ⊢ Rel 𝐴 |
relssi.2 | ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) |
Ref | Expression |
---|---|
relssi | ⊢ 𝐴 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relssi.1 | . . 3 ⊢ Rel 𝐴 | |
2 | ssrel 5795 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) |
4 | relssi.2 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) | |
5 | 4 | ax-gen 1792 | . 2 ⊢ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) |
6 | 3, 5 | mpgbir 1796 | 1 ⊢ 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 ∈ wcel 2106 ⊆ wss 3963 〈cop 4637 Rel wrel 5694 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: xpsspw 5822 oprssdm 7614 dftpos4 8269 enssdom 9016 idssen 9036 ttrcltr 9754 txuni2 23589 acycgr0v 35133 prclisacycgr 35136 txpss3v 35860 pprodss4v 35866 bj-idres 37143 xrnss3v 38354 aoprssdm 47152 |
Copyright terms: Public domain | W3C validator |