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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-elssuniab | GIF version |
Description: Version of elssuni 3608 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 2771 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
2 | elssuni 3608 | . 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 1393 {cab 2026 Ⅎwnfc 2165 [wsbc 2764 ⊆ wss 2917 ∪ cuni 3580 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sbc 2765 df-in 2924 df-ss 2931 df-uni 3581 |
This theorem is referenced by: (None) |
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