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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opreu2reu1 | Structured version Visualization version GIF version |
Description: Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.) |
Ref | Expression |
---|---|
opreu2reu1.a | ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒 ↔ 𝜑)) |
Ref | Expression |
---|---|
opreu2reu1 | ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2reu 32228 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) | |
2 | opreu2reu1.a | . . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒 ↔ 𝜑)) | |
3 | 2 | opreu2reurex 6287 | . 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜒 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) |
4 | 1, 3 | bitr4i 278 | 1 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wrex 3064 ∃!wreu 3368 ⟨cop 4629 × cxp 5667 ∃!w2reu 32227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-iun 4992 df-opab 5204 df-xp 5675 df-rel 5676 df-2reu 32228 |
This theorem is referenced by: (None) |
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