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Mirrors > Home > MPE Home > Th. List > ssrmof | 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 |
---|---|
ssrexf.1 | ⊢ Ⅎ𝑥𝐴 |
ssrexf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
ssrmof | ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrexf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
2 | ssrexf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | dfssf 3986 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 3 | biimpi 216 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
5 | pm3.45 622 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
6 | 5 | alimi 1808 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
7 | moim 2542 | . . 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 3378 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
10 | df-rmo 3378 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
11 | 8, 9, 10 | 3imtr4g 296 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∈ wcel 2106 ∃*wmo 2536 Ⅎwnfc 2888 ∃*wrmo 3377 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-mo 2538 df-clel 2814 df-nfc 2890 df-rmo 3378 df-ss 3980 |
This theorem is referenced by: 2sqreunnlem1 27508 2sqreunnlem2 27514 disjss1f 32592 upeu 48817 |
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