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Mirrors > Home > NFE Home > Th. List > ss2iun | GIF version |
Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ss2iun | ⊢ (∀x ∈ A B ⊆ C → ∪x ∈ A B ⊆ ∪x ∈ A C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3267 | . . . . 5 ⊢ (B ⊆ C → (y ∈ B → y ∈ C)) | |
2 | 1 | ralimi 2689 | . . . 4 ⊢ (∀x ∈ A B ⊆ C → ∀x ∈ A (y ∈ B → y ∈ C)) |
3 | rexim 2718 | . . . 4 ⊢ (∀x ∈ A (y ∈ B → y ∈ C) → (∃x ∈ A y ∈ B → ∃x ∈ A y ∈ C)) | |
4 | 2, 3 | syl 15 | . . 3 ⊢ (∀x ∈ A B ⊆ C → (∃x ∈ A y ∈ B → ∃x ∈ A y ∈ C)) |
5 | eliun 3973 | . . 3 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B) | |
6 | eliun 3973 | . . 3 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C) | |
7 | 4, 5, 6 | 3imtr4g 261 | . 2 ⊢ (∀x ∈ A B ⊆ C → (y ∈ ∪x ∈ A B → y ∈ ∪x ∈ A C)) |
8 | 7 | ssrdv 3278 | 1 ⊢ (∀x ∈ A B ⊆ C → ∪x ∈ A B ⊆ ∪x ∈ A C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 ⊆ wss 3257 ∪ciun 3969 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-iun 3971 |
This theorem is referenced by: iuneq2 3985 |
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