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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 | dfss2f 3951 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 3 | biimpi 218 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
5 | pm3.45 623 | . . . 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 2625 | . . 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 3145 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
10 | df-rmo 3145 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
11 | 8, 9, 10 | 3imtr4g 298 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1534 ∈ wcel 2113 ∃*wmo 2619 Ⅎwnfc 2960 ∃*wrmo 3140 ⊆ wss 3929 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rmo 3145 df-in 3936 df-ss 3945 |
This theorem is referenced by: 2sqreunnlem1 26021 2sqreunnlem2 26027 disjss1f 30320 |
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