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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 1923 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → 𝜒)) |
3 | ssopab2 5539 | . 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 3943 {copab 5203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-ss 3960 df-opab 5204 |
This theorem is referenced by: xpss12 5684 coss1 5849 coss2 5850 cnvss 5866 aceq3lem 10117 coss12d 14925 shftfval 15023 sslm 23158 ulmval 26271 mptssALT 32409 fpwrelmap 32465 cossss 37808 dicssdvh 40570 rfovcnvf1od 43331 |
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