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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 2831 | . 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 4805 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
5 | 4 | eqeq2i 2837 | . 2 ⊢ ({𝐶, 𝐷} = 〈𝐴, 𝐵〉 ↔ {𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}}) |
6 | opeqpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
7 | opeqpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
8 | snex 5335 | . . 3 ⊢ {𝐴} ∈ V | |
9 | prex 5336 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
10 | 6, 7, 8, 9 | preq12b 4784 | . 2 ⊢ ({𝐶, 𝐷} = {{𝐴}, {𝐴, 𝐵}} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
11 | 1, 5, 10 | 3bitri 299 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 {cpr 4572 〈cop 4576 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 |
This theorem is referenced by: propeqop 5400 relop 5724 |
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