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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 1931 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → 𝜒)) |
3 | ssopab2 5546 | . 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 1539 ⊆ wss 3948 {copab 5210 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3955 df-ss 3965 df-opab 5211 |
This theorem is referenced by: xpss12 5691 coss1 5855 coss2 5856 cnvss 5872 aceq3lem 10117 coss12d 14921 shftfval 15019 sslm 22810 ulmval 25899 mptssALT 31938 fpwrelmap 31996 cossss 37381 dicssdvh 40143 rfovcnvf1od 42837 |
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