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 2897 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ P ↔ 𝐴 ∈ P)) | |
2 | 1 | anbi1d 629 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝑦 ∈ P))) |
3 | psseq1 4061 | . . . 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 2897 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ P ↔ 𝐵 ∈ P)) | |
6 | 5 | anbi2d 628 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ P))) |
7 | psseq2 4062 | . . . 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 10395 | . . 3 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} | |
10 | 4, 8, 9 | brabg 5417 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵))) |
11 | 10 | bianabs 542 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊊ wpss 3934 class class class wbr 5057 Pcnp 10269 <P cltp 10273 |
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 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-ltp 10395 |
This theorem is referenced by: ltsopr 10442 ltaddpr 10444 ltexprlem7 10452 ltexpri 10453 suplem1pr 10462 suplem2pr 10463 |
Copyright terms: Public domain | W3C validator |