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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 1928 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → 𝜒)) |
| 3 | ssopab2 5551 | . 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 1538 ⊆ wss 3951 {copab 5205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ss 3968 df-opab 5206 |
| This theorem is referenced by: xpss12 5700 coss1 5866 coss2 5867 cnvss 5883 aceq3lem 10160 coss12d 15011 shftfval 15109 sslm 23307 ulmval 26423 mptssALT 32685 fpwrelmap 32744 cossss 38426 dicssdvh 41188 rfovcnvf1od 44017 |
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