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Mirrors > Home > MPE Home > Th. List > preqr1 | Structured version Visualization version GIF version |
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
preqr1.a | ⊢ 𝐴 ∈ V |
preqr1.b | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
preqr1 | ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preqr1.a | . . 3 ⊢ 𝐴 ∈ V | |
2 | id 22 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
3 | preqr1.b | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ V → 𝐵 ∈ V) |
5 | 2, 4 | preq1b 4769 | . . 3 ⊢ (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵)) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵) |
7 | 6 | biimpi 217 | 1 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {cpr 4559 |
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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-un 3938 df-sn 4558 df-pr 4560 |
This theorem is referenced by: preqr2 4772 opthwiener 5395 opthhausdorff0 5399 cusgrfilem2 27165 usgr2wlkneq 27464 wopprc 39505 |
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