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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 3625); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3779. (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 3061 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | nfv 1921 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
3 | rmo2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | 2, 3 | nfan 1906 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
5 | 4 | mof 2563 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)) |
6 | impexp 454 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | |
7 | 6 | albii 1826 | . . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) |
8 | df-ral 3058 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | |
9 | 7, 8 | bitr4i 281 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) |
10 | 9 | exbii 1854 | . 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) |
11 | 1, 5, 10 | 3bitri 300 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1540 ∃wex 1786 Ⅎwnf 1790 ∈ wcel 2114 ∃*wmo 2538 ∀wral 3053 ∃*wrmo 3056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-11 2162 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-mo 2540 df-ral 3058 df-rmo 3061 |
This theorem is referenced by: rmo2i 3779 rmoanimALT 3786 disjiun 5017 poimirlem2 35402 |
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