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Mirrors > Home > ILE Home > Th. List > ssopab2dv | 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 1875 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → 𝜒)) |
3 | ssopab2 4277 | . 2 ⊢ (∀𝑥∀𝑦(𝜓 → 𝜒) → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) | |
4 | 2, 3 | syl 14 | 1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 ⊆ wss 3131 {copab 4065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-in 3137 df-ss 3144 df-opab 4067 |
This theorem is referenced by: xpss12 4735 coss1 4784 coss2 4785 cnvss 4802 shftfvalg 10829 shftfval 10832 reldvdsrsrg 13266 dvdsrvald 13267 dvdsrex 13272 sslm 13832 |
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