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Mirrors > Home > ILE Home > Th. List > unssd | GIF version |
Description: A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
unssd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
unssd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
unssd | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unssd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
2 | unssd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | unss 3301 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
4 | 3 | biimpi 119 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
5 | 1, 2, 4 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∪ cun 3119 ⊆ wss 3121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 |
This theorem is referenced by: tpssi 3746 casef 7065 un0addcl 9168 un0mulcl 9169 fzosplit 10133 fzouzsplit 10135 exmidunben 12381 strleund 12506 fsumcncntop 13350 bj-charfun 13842 bj-omtrans 13991 |
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