![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3970 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 3 | biimpi 215 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
5 | pm3.45 621 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
6 | 5 | alimi 1806 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
7 | moim 2534 | . . 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 3373 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
10 | df-rmo 3373 | . 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 1532 ∈ wcel 2099 ∃*wmo 2528 Ⅎwnfc 2879 ∃*wrmo 3372 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-rmo 3373 df-v 3473 df-in 3954 df-ss 3964 |
This theorem is referenced by: 2sqreunnlem1 27395 2sqreunnlem2 27401 disjss1f 32375 |
Copyright terms: Public domain | W3C validator |