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Mirrors > Home > ILE Home > Th. List > ssrab3 | GIF version |
Description: Subclass relation for a restricted class abstraction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
ssrab3.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} |
Ref | Expression |
---|---|
ssrab3 | ⊢ 𝐵 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab3.1 | . 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} | |
2 | ssrab2 3240 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
3 | 1, 2 | eqsstri 3187 | 1 ⊢ 𝐵 ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 {crab 2459 ⊆ 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-rab 2464 df-in 3135 df-ss 3142 |
This theorem is referenced by: pcprecl 12259 pcprendvds 12260 lgsfcl2 14040 |
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