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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 3621 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | uneq1d 3303 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐴} ∪ {𝐶}) = ({𝐵} ∪ {𝐶})) |
3 | df-pr 3617 | . 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶}) | |
4 | df-pr 3617 | . 2 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶}) | |
5 | 2, 3, 4 | 3eqtr4g 2247 | 1 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∪ cun 3142 {csn 3610 {cpr 3611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3616 df-pr 3617 |
This theorem is referenced by: preq2 3688 preq12 3689 preq1i 3690 preq1d 3693 tpeq1 3696 prnzg 3734 preq12b 3788 preq12bg 3791 opeq1 3796 uniprg 3842 intprg 3895 prexg 4232 opthreg 4576 bdxmet 14486 bj-prexg 15149 |
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