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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 3974 | . . . 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 1811 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
| 7 | moim 2544 | . . 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 3380 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
| 10 | df-rmo 3380 | . 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 1538 ∈ wcel 2108 ∃*wmo 2538 Ⅎwnfc 2890 ∃*wrmo 3379 ⊆ wss 3951 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-mo 2540 df-clel 2816 df-nfc 2892 df-rmo 3380 df-ss 3968 |
| This theorem is referenced by: 2sqreunnlem1 27493 2sqreunnlem2 27499 disjss1f 32585 upeu 48928 |
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