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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 3334 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
4 | 3 | biimpi 120 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
5 | 1, 2, 4 | syl2anc 411 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∪ cun 3152 ⊆ wss 3154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-un 3158 df-in 3160 df-ss 3167 |
This theorem is referenced by: tpssi 3786 casef 7149 un0addcl 9276 un0mulcl 9277 fzosplit 10247 fzouzsplit 10249 4sqlem11 12542 4sqlem19 12550 exmidunben 12586 strleund 12724 lsptpcl 13893 lspun 13901 fsumcncntop 14746 plyf 14916 elplyr 14919 elplyd 14920 ply1term 14922 plyaddlem 14928 plymullem 14929 plycolemc 14936 plycjlemc 14938 plycj 14939 plycn 14940 bj-charfun 15369 bj-omtrans 15518 |
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