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| 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 5792 | . . 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 1795 | . 2 ⊢ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) | 
| 6 | 3, 5 | mpgbir 1799 | 1 ⊢ 𝐴 ⊆ 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 Rel wrel 5690 | 
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: xpsspw 5819 oprssdm 7614 dftpos4 8270 enssdom 9017 idssen 9037 ttrcltr 9756 txuni2 23573 acycgr0v 35153 prclisacycgr 35156 txpss3v 35879 pprodss4v 35885 bj-idres 37161 xrnss3v 38373 aoprssdm 47214 | 
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