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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 11017 | . 2 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} | |
2 | opabssxp 5765 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 4014 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∈ wcel 2099 ⊆ wss 3947 ⊊ wpss 3948 {copab 5206 × cxp 5671 Pcnp 10891 <P cltp 10895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-ss 3964 df-opab 5207 df-xp 5679 df-ltp 11017 |
This theorem is referenced by: ltexpri 11075 ltaprlem 11076 ltapr 11077 suplem1pr 11084 suplem2pr 11085 supexpr 11086 ltsrpr 11109 ltsosr 11126 mappsrpr 11140 |
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