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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 5126 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) | |
2 | 1 | unieqd 3832 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ {〈𝐵, 𝐴〉}) |
3 | elex 2760 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
4 | elex 2760 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
5 | opexg 4240 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → 〈𝐵, 𝐴〉 ∈ V) | |
6 | 3, 4, 5 | syl2anr 290 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐵, 𝐴〉 ∈ V) |
7 | unisng 3838 | . . 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 2220 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 Vcvv 2749 {csn 3604 〈cop 3607 ∪ cuni 3821 ◡ccnv 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-rel 4645 df-cnv 4646 |
This theorem is referenced by: 2nd1st 6194 cnvf1olem 6238 brtposg 6268 dftpos4 6277 tpostpos 6278 xpcomco 6839 fsumcnv 11458 fprodcnv 11646 txswaphmeolem 14091 |
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