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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 10260 | . 2 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} | |
2 | opabssxp 5536 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 3928 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2083 ⊆ wss 3865 ⊊ wpss 3866 {copab 5030 × cxp 5448 Pcnp 10134 <P cltp 10138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-in 3872 df-ss 3880 df-opab 5031 df-xp 5456 df-ltp 10260 |
This theorem is referenced by: ltexpri 10318 ltaprlem 10319 ltapr 10320 suplem1pr 10327 suplem2pr 10328 supexpr 10329 ltsrpr 10352 ltsosr 10369 mappsrpr 10383 |
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