Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > opswapg | GIF version |
Description: Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Ref | Expression |
---|---|
opswapg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvsng 5019 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) | |
2 | 1 | unieqd 3742 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ {〈𝐵, 𝐴〉}) |
3 | elex 2692 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
4 | elex 2692 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
5 | opexg 4145 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → 〈𝐵, 𝐴〉 ∈ V) | |
6 | 3, 4, 5 | syl2anr 288 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐵, 𝐴〉 ∈ V) |
7 | unisng 3748 | . . 3 ⊢ (〈𝐵, 𝐴〉 ∈ V → ∪ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉) | |
8 | 6, 7 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉) |
9 | 2, 8 | eqtrd 2170 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2681 {csn 3522 〈cop 3525 ∪ cuni 3731 ◡ccnv 4533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 |
This theorem is referenced by: 2nd1st 6071 cnvf1olem 6114 brtposg 6144 dftpos4 6153 tpostpos 6154 xpcomco 6713 fsumcnv 11199 txswaphmeolem 12478 |
Copyright terms: Public domain | W3C validator |