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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2exopprim | Structured version Visualization version GIF version |
Description: The existence of an ordered pair fulfilling a wff implies the existence of an unordered pair fulfilling the wff. (Contributed by AV, 29-Jul-2023.) |
Ref | Expression |
---|---|
2exopprim | ⊢ (∃𝑎∃𝑏(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppr 43314 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝑎, 𝑏} = {𝐴, 𝐵})) | |
2 | 1 | el2v 3501 | . . . . 5 ⊢ (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝑎, 𝑏} = {𝐴, 𝐵}) |
3 | 2 | eqcomd 2827 | . . . 4 ⊢ (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝐴, 𝐵} = {𝑎, 𝑏}) |
4 | 3 | eqcoms 2829 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 → {𝐴, 𝐵} = {𝑎, 𝑏}) |
5 | 4 | anim1i 616 | . 2 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)) |
6 | 5 | 2eximi 1836 | 1 ⊢ (∃𝑎∃𝑏(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 Vcvv 3494 {cpr 4569 〈cop 4573 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 |
This theorem is referenced by: (None) |
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