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Mirrors > Home > MPE Home > Th. List > eqbrrdv | Structured version Visualization version GIF version |
Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
eqbrrdv.1 | ⊢ (𝜑 → Rel 𝐴) |
eqbrrdv.2 | ⊢ (𝜑 → Rel 𝐵) |
eqbrrdv.3 | ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
Ref | Expression |
---|---|
eqbrrdv | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrrdv.3 | . . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
2 | df-br 5058 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | df-br 5058 | . . . 4 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
4 | 1, 2, 3 | 3bitr3g 314 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
5 | 4 | alrimivv 1920 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | eqbrrdv.1 | . . 3 ⊢ (𝜑 → Rel 𝐴) | |
7 | eqbrrdv.2 | . . 3 ⊢ (𝜑 → Rel 𝐵) | |
8 | eqrel 5651 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
9 | 6, 7, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) |
10 | 5, 9 | mpbird 258 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 = wceq 1528 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-in 3940 df-ss 3949 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: eqbrrdva 5733 oppcsect2 17037 qusxpid 30855 erim2 35791 |
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