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Mirrors > Home > MPE Home > Th. List > relssdv | Structured version Visualization version GIF version |
Description: Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.) |
Ref | Expression |
---|---|
relssdv.1 | ⊢ (𝜑 → Rel 𝐴) |
relssdv.2 | ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) |
Ref | Expression |
---|---|
relssdv | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relssdv.2 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | 1 | alrimivv 1929 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) |
3 | relssdv.1 | . . 3 ⊢ (𝜑 → Rel 𝐴) | |
4 | ssrel 5621 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) |
6 | 2, 5 | mpbird 260 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∈ wcel 2111 ⊆ wss 3881 〈cop 4531 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: relssres 5859 poirr2 5951 sofld 6011 relssdmrn 6088 funcres2 17160 wunfunc 17161 fthres2 17194 pospo 17575 joindmss 17609 meetdmss 17623 clatl 17718 subrgdvds 19542 opsrtoslem2 20724 txcls 22209 txdis1cn 22240 txkgen 22257 qustgplem 22726 metustid 23161 metustexhalf 23163 ovoliunlem1 24106 dvres2 24515 cvmlift2lem12 32674 dib2dim 38539 dih2dimbALTN 38541 dihmeetlem1N 38586 dihglblem5apreN 38587 dihmeetlem13N 38615 dihjatcclem4 38717 |
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