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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 2629 | . 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 4401 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
5 | 4 | eqeq2i 2634 | . 2 ⊢ ({𝐶, 𝐷} = 〈𝐴, 𝐵〉 ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}}) |
6 | opeqpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
7 | opeqpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
8 | snex 4908 | . . 3 ⊢ {𝐴} ∈ V | |
9 | prex 4909 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
10 | 6, 7, 8, 9 | preq12b 4382 | . 2 ⊢ ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
11 | 1, 5, 10 | 3bitri 286 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 {cpr 4179 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: propeqop 4970 relop 5272 |
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