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Mirrors > Home > MPE Home > Th. List > ssuniOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of ssuni 4459 as of 26-Jul-2021. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ssuniOLD | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2690 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐵)) | |
2 | 1 | imbi1d 331 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝐶))) |
3 | elunii 4441 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶) | |
4 | 3 | expcom 451 | . . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝐶)) |
5 | 2, 4 | vtoclga 3272 | . . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝐶)) |
6 | 5 | imim2d 57 | . . . 4 ⊢ (𝐵 ∈ 𝐶 → ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶))) |
7 | 6 | alimdv 1845 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶))) |
8 | dfss2 3591 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵)) | |
9 | dfss2 3591 | . . 3 ⊢ (𝐴 ⊆ ∪ 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶)) | |
10 | 7, 8, 9 | 3imtr4g 285 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊆ ∪ 𝐶)) |
11 | 10 | impcom 446 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∪ cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-uni 4437 |
This theorem is referenced by: (None) |
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