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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 2733 | . 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 4867 | . . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} |
5 | 4 | eqeq2i 2739 | . 2 ⊢ ({𝐶, 𝐷} = ⟨𝐴, 𝐵⟩ ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}}) |
6 | opeqpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
7 | opeqpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
8 | snex 5424 | . . 3 ⊢ {𝐴} ∈ V | |
9 | prex 5425 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
10 | 6, 7, 8, 9 | preq12b 4846 | . 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 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 {cpr 4625 ⟨cop 4629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 |
This theorem is referenced by: propeqop 5500 relop 5843 |
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