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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-elssuniab | GIF version |
Description: Version of elssuni 3599 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.) |
Ref | Expression |
---|---|
bj-elssuniab.nf | ⊢ ℲxA |
Ref | Expression |
---|---|
bj-elssuniab | ⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc8g 2765 | . 2 ⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ})) | |
2 | elssuni 3599 | . 2 ⊢ (A ∈ {x ∣ φ} → A ⊆ ∪ {x ∣ φ}) | |
3 | 1, 2 | syl6bi 152 | 1 ⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1390 {cab 2023 Ⅎwnfc 2162 [wsbc 2758 ⊆ wss 2911 ∪ cuni 3571 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-sbc 2759 df-in 2918 df-ss 2925 df-uni 3572 |
This theorem is referenced by: (None) |
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