![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ltprord | Structured version Visualization version GIF version |
Description: Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltprord | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2816 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ P ↔ 𝐴 ∈ P)) | |
2 | 1 | anbi1d 629 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝑦 ∈ P))) |
3 | psseq1 4083 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦)) | |
4 | 2, 3 | anbi12d 630 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦) ↔ ((𝐴 ∈ P ∧ 𝑦 ∈ P) ∧ 𝐴 ⊊ 𝑦))) |
5 | eleq1 2816 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ P ↔ 𝐵 ∈ P)) | |
6 | 5 | anbi2d 628 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ P))) |
7 | psseq2 4084 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵)) | |
8 | 6, 7 | anbi12d 630 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ P ∧ 𝑦 ∈ P) ∧ 𝐴 ⊊ 𝑦) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵))) |
9 | df-ltp 11000 | . . 3 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} | |
10 | 4, 8, 9 | brabg 5535 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵))) |
11 | 10 | bianabs 541 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊊ wpss 3945 class class class wbr 5142 Pcnp 10874 <P cltp 10878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-ltp 11000 |
This theorem is referenced by: ltsopr 11047 ltaddpr 11049 ltexprlem7 11057 ltexpri 11058 suplem1pr 11067 suplem2pr 11068 |
Copyright terms: Public domain | W3C validator |