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Mirrors > Home > ILE Home > Th. List > abssi | GIF version |
Description: Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) |
Ref | Expression |
---|---|
abssi.1 | ⊢ (𝜑 → 𝑥 ∈ 𝐴) |
Ref | Expression |
---|---|
abssi | ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abssi.1 | . . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
2 | 1 | ss2abi 3227 | . 2 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} |
3 | abid2 2298 | . 2 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
4 | 2, 3 | sseqtri 3189 | 1 ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 {cab 2163 ⊆ wss 3129 |
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 3135 df-ss 3142 |
This theorem is referenced by: ssab2 3239 abf 3466 intab 3873 opabss 4067 relopabi 4752 exse2 5002 mpoexw 6213 tfrlem8 6318 frecabex 6398 fiprc 6814 fival 6968 nqprxx 7544 ltnqex 7547 gtnqex 7548 recexprlemell 7620 recexprlemelu 7621 recexprlempr 7630 4sqlem1 12380 topnex 13517 2sqlem7 14388 |
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