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Mirrors > Home > ILE Home > Th. List > erth2 | GIF version |
Description: Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) |
Ref | Expression |
---|---|
erth2.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
erth2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
Ref | Expression |
---|---|
erth2 | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erth2.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
2 | 1 | ersymb 6436 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
3 | erth2.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
4 | 1, 3 | erth 6466 | . . 3 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐵]𝑅 = [𝐴]𝑅)) |
5 | eqcom 2139 | . . 3 ⊢ ([𝐵]𝑅 = [𝐴]𝑅 ↔ [𝐴]𝑅 = [𝐵]𝑅) | |
6 | 4, 5 | syl6bb 195 | . 2 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
7 | 2, 6 | bitrd 187 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 Er wer 6419 [cec 6420 |
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-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-er 6422 df-ec 6424 |
This theorem is referenced by: qliftel 6502 |
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