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 30238 | . 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 6138 | . 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜒 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) |
4 | 1, 3 | bitr4i 280 | 1 ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wrex 3138 ∃!wreu 3139 〈cop 4566 × cxp 5546 ∃!w2reu 30237 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-iun 4914 df-opab 5122 df-xp 5554 df-rel 5555 df-2reu 30238 |
This theorem is referenced by: (None) |
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