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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 10564 | . 2 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} | |
2 | opabssxp 5625 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 3921 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2112 ⊆ wss 3853 ⊊ wpss 3854 {copab 5101 × cxp 5534 Pcnp 10438 <P cltp 10442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-in 3860 df-ss 3870 df-opab 5102 df-xp 5542 df-ltp 10564 |
This theorem is referenced by: ltexpri 10622 ltaprlem 10623 ltapr 10624 suplem1pr 10631 suplem2pr 10632 supexpr 10633 ltsrpr 10656 ltsosr 10673 mappsrpr 10687 |
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