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Mirrors > Home > MPE Home > Th. List > opeqpr | Structured version Visualization version GIF version |
Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.) |
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 2825 | . 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 4794 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
5 | 4 | eqeq2i 2831 | . 2 ⊢ ({𝐶, 𝐷} = 〈𝐴, 𝐵〉 ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}}) |
6 | opeqpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
7 | opeqpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
8 | snex 5322 | . . 3 ⊢ {𝐴} ∈ V | |
9 | prex 5323 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
10 | 6, 7, 8, 9 | preq12b 4773 | . 2 ⊢ ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
11 | 1, 5, 10 | 3bitri 298 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 {cpr 4559 〈cop 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 |
This theorem is referenced by: propeqop 5388 relop 5714 |
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