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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2reu4 | Structured version Visualization version GIF version |
Description: Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2736. (Contributed by Alexander van der Vekens, 1-Jul-2017.) |
Ref | Expression |
---|---|
2reu4 | ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reurex 3372 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | |
2 | rexn0 4296 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ≠ ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ≠ ∅) |
4 | reurex 3372 | . . . 4 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) | |
5 | rexn0 4296 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → 𝐵 ≠ ∅) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → 𝐵 ≠ ∅) |
7 | 3, 6 | anim12i 606 | . 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
8 | ne0i 4150 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
9 | ne0i 4150 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → 𝐵 ≠ ∅) | |
10 | 8, 9 | anim12i 606 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
11 | 10 | a1d 25 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅))) |
12 | 11 | rexlimivv 3246 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
13 | 12 | adantr 474 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
14 | 2reu4a 42007 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))) | |
15 | 7, 13, 14 | pm5.21nii 370 | 1 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 ∃!wreu 3119 ∅c0 4144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-v 3416 df-dif 3801 df-nul 4145 |
This theorem is referenced by: (None) |
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