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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 3057 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵)) | |
2 | 1 | anim2d 333 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))) |
3 | 2 | eximdv 1834 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))) |
4 | eluni 3705 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) | |
5 | eluni 3705 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) | |
6 | 3, 4, 5 | 3imtr4g 204 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ ∪ 𝐵)) |
7 | 6 | ssrdv 3069 | 1 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1451 ∈ wcel 1463 ⊆ wss 3037 ∪ cuni 3702 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-in 3043 df-ss 3050 df-uni 3703 |
This theorem is referenced by: unissi 3725 unissd 3726 intssuni2m 3761 relfld 5025 tgcl 12076 distop 12097 |
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