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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 47042 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝑎, 𝑏} = {𝐴, 𝐵})) | |
| 2 | 1 | el2v 3487 | . . . . 5 ⊢ (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝑎, 𝑏} = {𝐴, 𝐵}) |
| 3 | 2 | eqcomd 2743 | . . . 4 ⊢ (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝐴, 𝐵} = {𝑎, 𝑏}) |
| 4 | 3 | eqcoms 2745 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 → {𝐴, 𝐵} = {𝑎, 𝑏}) |
| 5 | 4 | anim1i 615 | . 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 395 = wceq 1540 ∃wex 1779 Vcvv 3480 {cpr 4628 〈cop 4632 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: (None) |
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