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Mirrors > Home > MPE Home > Th. List > ssun3 | Structured version Visualization version GIF version |
Description: Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ssun3 | ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4172 | . 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) | |
2 | sstr2 3987 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐵 ∪ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))) | |
3 | 1, 2 | mpi 20 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3945 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-un 3952 df-in 3954 df-ss 3964 |
This theorem is referenced by: ssun 4189 ssunsn2 4831 xpsspw 5811 wfrlem15OLD 8344 uncmp 23320 alexsubALTlem3 23966 sxbrsigalem0 33891 bnj1450 34681 fineqvac 34717 altxpsspw 35573 pibt2 36896 superuncl 42998 cnvtrcl0 43056 |
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