![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ssopab2i | GIF version |
Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
Ref | Expression |
---|---|
ssopab2i.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
ssopab2i | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssopab2 4205 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) | |
2 | ssopab2i.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
3 | 2 | ax-gen 1426 | . 2 ⊢ ∀𝑦(𝜑 → 𝜓) |
4 | 1, 3 | mpg 1428 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 ⊆ wss 3076 {copab 3996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-in 3082 df-ss 3089 df-opab 3998 |
This theorem is referenced by: brab2a 4600 opabssxp 4621 funopab4 5168 ssoprab2i 5868 npsspw 7303 |
Copyright terms: Public domain | W3C validator |