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Mirrors > Home > NFE Home > Th. List > uniss2 | 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 4011 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 3913 | . . . . 5 ⊢ ((x ⊆ y ∧ y ∈ B) → x ⊆ ∪B) | |
2 | 1 | expcom 424 | . . . 4 ⊢ (y ∈ B → (x ⊆ y → x ⊆ ∪B)) |
3 | 2 | rexlimiv 2732 | . . 3 ⊢ (∃y ∈ B x ⊆ y → x ⊆ ∪B) |
4 | 3 | ralimi 2689 | . 2 ⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∀x ∈ A x ⊆ ∪B) |
5 | unissb 3921 | . 2 ⊢ (∪A ⊆ ∪B ↔ ∀x ∈ A x ⊆ ∪B) | |
6 | 4, 5 | sylibr 203 | 1 ⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∪A ⊆ ∪B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 ⊆ wss 3257 ∪cuni 3891 |
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-uni 3892 |
This theorem is referenced by: unidif 3923 |
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