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Mirrors > Home > MPE Home > Th. List > 2reu4 | Structured version Visualization version GIF version |
Description: Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2649. (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 3379 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | |
2 | rexn0 4510 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ≠ ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ≠ ∅) |
4 | reurex 3379 | . . . 4 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) | |
5 | rexn0 4510 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → 𝐵 ≠ ∅) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → 𝐵 ≠ ∅) |
7 | 3, 6 | anim12i 612 | . 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
8 | ne0i 4334 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
9 | ne0i 4334 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → 𝐵 ≠ ∅) | |
10 | 8, 9 | anim12i 612 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
11 | 10 | a1d 25 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅))) |
12 | 11 | rexlimivv 3198 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
13 | 12 | adantr 480 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
14 | 2reu4lem 4525 | . 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 378 | 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 205 ∧ wa 395 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∃!wreu 3373 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-dif 3951 df-nul 4323 |
This theorem is referenced by: opreu2reurex 6293 opreu2reuALT 32152 |
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