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Mirrors > Home > MPE Home > Th. List > ssopab2dv | Structured version Visualization version GIF version |
Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
ssopab2dv.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ssopab2dv | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssopab2dv.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | alrimivv 1925 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → 𝜒)) |
3 | ssopab2 5432 | . 2 ⊢ (∀𝑥∀𝑦(𝜓 → 𝜒) → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜒}) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ⊆ wss 3935 {copab 5127 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3942 df-ss 3951 df-opab 5128 |
This theorem is referenced by: xpss12 5569 coss1 5725 coss2 5726 cnvss 5742 aceq3lem 9545 coss12d 14331 shftfval 14428 sslm 21906 ulmval 24967 mptssALT 30419 fpwrelmap 30468 cossss 35669 dicssdvh 38321 rfovcnvf1od 40348 |
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