Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssrmo | Structured version Visualization version GIF version |
Description: "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.) |
Ref | Expression |
---|---|
ssrmo.1 | ⊢ Ⅎ𝑥𝐴 |
ssrmo.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
ssrmo | ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrmo.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
2 | ssrmo.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | dfss2f 3559 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 3 | biimpi 205 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
5 | pm3.45 875 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
6 | 5 | alimi 1730 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
7 | moim 2507 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)) → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
8 | 4, 6, 7 | 3syl 18 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
9 | df-rmo 2904 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
10 | df-rmo 2904 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
11 | 8, 9, 10 | 3imtr4g 284 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 ∃*wmo 2459 Ⅎwnfc 2738 ∃*wrmo 2899 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rmo 2904 df-in 3547 df-ss 3554 |
This theorem is referenced by: disjss1f 28768 |
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