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Mirrors > Home > MPE Home > Th. List > ltrelpr | Structured version Visualization version GIF version |
Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltrelpr | ⊢ <P ⊆ (P × P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ltp 10395 | . 2 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} | |
2 | opabssxp 5636 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 3998 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 ⊊ wpss 3934 {copab 5119 × cxp 5546 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 |
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-nfc 2960 df-in 3940 df-ss 3949 df-opab 5120 df-xp 5554 df-ltp 10395 |
This theorem is referenced by: ltexpri 10453 ltaprlem 10454 ltapr 10455 suplem1pr 10462 suplem2pr 10463 supexpr 10464 ltsrpr 10487 ltsosr 10504 mappsrpr 10518 |
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