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Mirrors > Home > ILE Home > Th. List > preq1 | GIF version |
Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
Ref | Expression |
---|---|
preq1 | ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3538 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | uneq1d 3229 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐴} ∪ {𝐶}) = ({𝐵} ∪ {𝐶})) |
3 | df-pr 3534 | . 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶}) | |
4 | df-pr 3534 | . 2 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶}) | |
5 | 2, 3, 4 | 3eqtr4g 2197 | 1 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∪ cun 3069 {csn 3527 {cpr 3528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 |
This theorem is referenced by: preq2 3601 preq12 3602 preq1i 3603 preq1d 3606 tpeq1 3609 prnzg 3647 preq12b 3697 preq12bg 3700 opeq1 3705 uniprg 3751 intprg 3804 prexg 4133 opthreg 4471 bdxmet 12670 bj-prexg 13109 |
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