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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 2820 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ P ↔ 𝐴 ∈ P)) | |
2 | 1 | anbi1d 633 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝑦 ∈ P))) |
3 | psseq1 3978 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦)) | |
4 | 2, 3 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦) ↔ ((𝐴 ∈ P ∧ 𝑦 ∈ P) ∧ 𝐴 ⊊ 𝑦))) |
5 | eleq1 2820 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ P ↔ 𝐵 ∈ P)) | |
6 | 5 | anbi2d 632 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ P))) |
7 | psseq2 3979 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵)) | |
8 | 6, 7 | anbi12d 634 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ P ∧ 𝑦 ∈ P) ∧ 𝐴 ⊊ 𝑦) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵))) |
9 | df-ltp 10485 | . . 3 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} | |
10 | 4, 8, 9 | brabg 5394 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵))) |
11 | 10 | bianabs 545 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊊ wpss 3844 class class class wbr 5030 Pcnp 10359 <P cltp 10363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-ltp 10485 |
This theorem is referenced by: ltsopr 10532 ltaddpr 10534 ltexprlem7 10542 ltexpri 10543 suplem1pr 10552 suplem2pr 10553 |
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