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Mirrors > Home > ILE Home > Th. List > uniss | GIF version |
Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
uniss | ⊢ (A ⊆ B → ∪ A ⊆ ∪ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2933 | . . . . 5 ⊢ (A ⊆ B → (y ∈ A → y ∈ B)) | |
2 | 1 | anim2d 320 | . . . 4 ⊢ (A ⊆ B → ((x ∈ y ∧ y ∈ A) → (x ∈ y ∧ y ∈ B))) |
3 | 2 | eximdv 1757 | . . 3 ⊢ (A ⊆ B → (∃y(x ∈ y ∧ y ∈ A) → ∃y(x ∈ y ∧ y ∈ B))) |
4 | eluni 3574 | . . 3 ⊢ (x ∈ ∪ A ↔ ∃y(x ∈ y ∧ y ∈ A)) | |
5 | eluni 3574 | . . 3 ⊢ (x ∈ ∪ B ↔ ∃y(x ∈ y ∧ y ∈ B)) | |
6 | 3, 4, 5 | 3imtr4g 194 | . 2 ⊢ (A ⊆ B → (x ∈ ∪ A → x ∈ ∪ B)) |
7 | 6 | ssrdv 2945 | 1 ⊢ (A ⊆ B → ∪ A ⊆ ∪ B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃wex 1378 ∈ wcel 1390 ⊆ 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-in 2918 df-ss 2925 df-uni 3572 |
This theorem is referenced by: unissi 3594 unissd 3595 intssuni2m 3630 relfld 4789 |
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