![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eqrelriv | Structured version Visualization version GIF version |
Description: Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.) |
Ref | Expression |
---|---|
eqrelriv.1 | ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) |
Ref | Expression |
---|---|
eqrelriv | ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelriv.1 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) | |
2 | 1 | gen2 1799 | . 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) |
3 | eqrel 5745 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))) | |
4 | 2, 3 | mpbiri 258 | 1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ⟨cop 4597 Rel wrel 5643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-in 3922 df-ss 3932 df-opab 5173 df-xp 5644 df-rel 5645 |
This theorem is referenced by: eqrelriiv 5751 dfrel2 6146 coi1 6219 cnviin 6243 |
Copyright terms: Public domain | W3C validator |