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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 2744 | . 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 4872 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
| 5 | 4 | eqeq2i 2750 | . 2 ⊢ ({𝐶, 𝐷} = 〈𝐴, 𝐵〉 ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}}) |
| 6 | opeqpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
| 7 | opeqpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
| 8 | snex 5436 | . . 3 ⊢ {𝐴} ∈ V | |
| 9 | prex 5437 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
| 10 | 6, 7, 8, 9 | preq12b 4850 | . 2 ⊢ ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
| 11 | 1, 5, 10 | 3bitri 297 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 {cpr 4628 〈cop 4632 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: propeqop 5512 relop 5861 |
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