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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 5207 | . . 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 1722 | . 2 ⊢ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) |
6 | 3, 5 | mpgbir 1726 | 1 ⊢ 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 ∈ wcel 1990 ⊆ wss 3574 〈cop 4183 Rel wrel 5119 |
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-in 3581 df-ss 3588 df-opab 4713 df-xp 5120 df-rel 5121 |
This theorem is referenced by: xpsspw 5233 oprssdm 6815 resiexg 7102 dftpos4 7371 enssdom 7980 idssen 8000 txuni2 21368 txpss3v 31985 pprodss4v 31991 xrnss3v 34135 aoprssdm 41282 |
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