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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 4627 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵) |
5 | 1, 2, 4 | mp2an 422 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 〈cop 3525 Rel wrel 4539 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-xp 4540 df-rel 4541 |
This theorem is referenced by: eqbrriv 4629 inopab 4666 difopab 4667 dfres2 4866 cnvopab 4935 cnv0 4937 cnvdif 4940 cnvcnvsn 5010 dfco2 5033 coiun 5043 co02 5047 coass 5052 ressn 5074 |
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