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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 1843 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → 𝜒)) |
3 | ssopab2 4926 | . 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 1473 ⊆ wss 3540 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 df-opab 4644 |
This theorem is referenced by: xpss12 5148 coss1 5199 coss2 5200 cnvss 5216 cnvssOLD 5217 aceq3lem 8826 coss12d 13559 shftfval 13658 sslm 20913 ulmval 23938 clwlkswlks 26286 iseupa 26492 mptssALT 28857 fpwrelmap 28896 dicssdvh 35493 rfovcnvf1od 37318 |
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