Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rmo2 | Structured version Visualization version GIF version |
Description: Alternate definition of restricted "at most one." Note that ∃*𝑥 ∈ 𝐴𝜑 is not equivalent to ∃𝑦 ∈ 𝐴∀𝑥 ∈ 𝐴(𝜑 → 𝑥 = 𝑦) (in analogy to reu6 3714); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3868. (Contributed by NM, 17-Jun-2017.) |
Ref | Expression |
---|---|
rmo2.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
rmo2 | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 3143 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
3 | rmo2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | 2, 3 | nfan 1891 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
5 | 4 | mof 2640 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)) |
6 | impexp 451 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | |
7 | 6 | albii 1811 | . . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) |
8 | df-ral 3140 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | |
9 | 7, 8 | bitr4i 279 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) |
10 | 9 | exbii 1839 | . 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) |
11 | 1, 5, 10 | 3bitri 298 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 ∈ wcel 2105 ∃*wmo 2613 ∀wral 3135 ∃*wrmo 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-mo 2615 df-ral 3140 df-rmo 3143 |
This theorem is referenced by: rmo2i 3868 rmoanimALT 3876 disjiun 5044 poimirlem2 34775 |
Copyright terms: Public domain | W3C validator |