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Mathbox for Alexander van der Vekens |
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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 46980 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝑎, 𝑏} = {𝐴, 𝐵})) | |
2 | 1 | el2v 3485 | . . . . 5 ⊢ (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝑎, 𝑏} = {𝐴, 𝐵}) |
3 | 2 | eqcomd 2741 | . . . 4 ⊢ (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝐴, 𝐵} = {𝑎, 𝑏}) |
4 | 3 | eqcoms 2743 | . . 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 1833 | 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 1537 ∃wex 1776 Vcvv 3478 {cpr 4633 〈cop 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: (None) |
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