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Mirrors > Home > ILE Home > Th. List > opeqpr | GIF version |
Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) |
Ref | Expression |
---|---|
opeqpr.1 | ⊢ 𝐴 ∈ V |
opeqpr.2 | ⊢ 𝐵 ∈ V |
opeqpr.3 | ⊢ 𝐶 ∈ V |
opeqpr.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
opeqpr | ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2139 | . 2 ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ {𝐶, 𝐷} = 〈𝐴, 𝐵〉) | |
2 | opeqpr.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | opeqpr.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | dfop 3699 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
5 | 4 | eqeq2i 2148 | . 2 ⊢ ({𝐶, 𝐷} = 〈𝐴, 𝐵〉 ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}}) |
6 | opeqpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
7 | opeqpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
8 | 2 | snex 4104 | . . 3 ⊢ {𝐴} ∈ V |
9 | prexg 4128 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
10 | 2, 3, 9 | mp2an 422 | . . 3 ⊢ {𝐴, 𝐵} ∈ V |
11 | 6, 7, 8, 10 | preq12b 3692 | . 2 ⊢ ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
12 | 1, 5, 11 | 3bitri 205 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 Vcvv 2681 {csn 3522 {cpr 3523 〈cop 3525 |
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-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 |
This theorem is referenced by: relop 4684 |
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