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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pairreueq | Structured version Visualization version GIF version | ||
| Description: Two equivalent representations of the existence of a unique proper pair. (Contributed by AV, 1-Mar-2023.) |
| Ref | Expression |
|---|---|
| pairreueq.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
| Ref | Expression |
|---|---|
| pairreueq | ⊢ (∃!𝑝 ∈ 𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveqeq2 6915 | . . . . . 6 ⊢ (𝑥 = 𝑝 → ((♯‘𝑥) = 2 ↔ (♯‘𝑝) = 2)) | |
| 2 | pairreueq.p | . . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} | |
| 3 | 1, 2 | elrab2 3695 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2)) |
| 4 | 3 | anbi1i 624 | . . . 4 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝜑) ↔ ((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑)) |
| 5 | anass 468 | . . . 4 ⊢ (((𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2) ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑))) | |
| 6 | 4, 5 | bitri 275 | . . 3 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝜑) ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑))) |
| 7 | 6 | eubii 2585 | . 2 ⊢ (∃!𝑝(𝑝 ∈ 𝑃 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑))) |
| 8 | df-reu 3381 | . 2 ⊢ (∃!𝑝 ∈ 𝑃 𝜑 ↔ ∃!𝑝(𝑝 ∈ 𝑃 ∧ 𝜑)) | |
| 9 | df-reu 3381 | . 2 ⊢ (∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑) ↔ ∃!𝑝(𝑝 ∈ 𝒫 𝑉 ∧ ((♯‘𝑝) = 2 ∧ 𝜑))) | |
| 10 | 7, 8, 9 | 3bitr4i 303 | 1 ⊢ (∃!𝑝 ∈ 𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!weu 2568 ∃!wreu 3378 {crab 3436 𝒫 cpw 4600 ‘cfv 6561 2c2 12321 ♯chash 14369 |
| 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-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: requad2 47610 |
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