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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 10009 | . 2 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} | |
2 | opabssxp 5333 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 3784 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 ∈ wcel 2145 ⊆ wss 3723 ⊊ wpss 3724 {copab 4846 × cxp 5247 Pcnp 9883 <P cltp 9887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-in 3730 df-ss 3737 df-opab 4847 df-xp 5255 df-ltp 10009 |
This theorem is referenced by: ltexpri 10067 ltaprlem 10068 ltapr 10069 suplem1pr 10076 suplem2pr 10077 supexpr 10078 ltsrpr 10100 ltsosr 10117 mappsrpr 10131 |
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