Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 43622 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝑎, 𝑏} = {𝐴, 𝐵})) | |
2 | 1 | el2v 3448 | . . . . 5 ⊢ (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝑎, 𝑏} = {𝐴, 𝐵}) |
3 | 2 | eqcomd 2804 | . . . 4 ⊢ (〈𝑎, 𝑏〉 = 〈𝐴, 𝐵〉 → {𝐴, 𝐵} = {𝑎, 𝑏}) |
4 | 3 | eqcoms 2806 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 → {𝐴, 𝐵} = {𝑎, 𝑏}) |
5 | 4 | anim1i 617 | . 2 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)) |
6 | 5 | 2eximi 1837 | 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 399 = wceq 1538 ∃wex 1781 Vcvv 3441 {cpr 4527 〈cop 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |