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Mirrors > Home > ILE Home > Th. List > eqrelriiv | GIF version |
Description: Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.) |
Ref | Expression |
---|---|
eqreliiv.1 | ⊢ Rel 𝐴 |
eqreliiv.2 | ⊢ Rel 𝐵 |
eqreliiv.3 | ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) |
Ref | Expression |
---|---|
eqrelriiv | ⊢ 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqreliiv.1 | . 2 ⊢ Rel 𝐴 | |
2 | eqreliiv.2 | . 2 ⊢ Rel 𝐵 | |
3 | eqreliiv.3 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
4 | 3 | eqrelriv 4737 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵) |
5 | 1, 2, 4 | mp2an 426 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∈ wcel 2160 〈cop 3610 Rel wrel 4649 |
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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 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-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-opab 4080 df-xp 4650 df-rel 4651 |
This theorem is referenced by: eqbrriv 4739 inopab 4777 difopab 4778 dfres2 4977 restidsing 4981 cnvopab 5048 cnv0 5050 cnvdif 5053 cnvcnvsn 5123 dfco2 5146 coiun 5156 co02 5160 coass 5165 ressn 5187 |
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