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Mirrors > Home > MPE Home > Th. List > uniss2 | Structured version Visualization version GIF version |
Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 5051 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.) |
Ref | Expression |
---|---|
uniss2 | ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssuni 4935 | . . . . 5 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ⊆ ∪ 𝐵) | |
2 | 1 | expcom 412 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵)) |
3 | 2 | rexlimiv 3146 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵) |
4 | 3 | ralimi 3081 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵) |
5 | unissb 4942 | . 2 ⊢ (∪ 𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 ⊆ wss 3947 ∪ cuni 4907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-v 3474 df-in 3954 df-ss 3964 df-uni 4908 |
This theorem is referenced by: unidif 4945 coflim 10258 onsssupeqcond 42332 |
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