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Mirrors > Home > ILE Home > Th. List > ltrelpr | GIF version |
Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) |
Ref | Expression |
---|---|
ltrelpr | ⊢ <P ⊆ (P × P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iltp 6799 | . 2 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} | |
2 | opabssxp 4461 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 3039 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ∈ wcel 1434 ∃wrex 2354 ⊆ wss 2983 {copab 3859 × cxp 4390 ‘cfv 4953 1st c1st 5818 2nd c2nd 5819 Qcnq 6609 Pcnp 6620 <P cltp 6624 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-in 2989 df-ss 2996 df-opab 3861 df-xp 4398 df-iltp 6799 |
This theorem is referenced by: ltprordil 6918 ltexprlemm 6929 ltexprlemopl 6930 ltexprlemlol 6931 ltexprlemopu 6932 ltexprlemupu 6933 ltexprlemdisj 6935 ltexprlemloc 6936 ltexprlemfl 6938 ltexprlemrl 6939 ltexprlemfu 6940 ltexprlemru 6941 ltexpri 6942 lteupri 6946 ltaprlem 6947 prplnqu 6949 caucvgprprlemk 7012 caucvgprprlemnkltj 7018 caucvgprprlemnkeqj 7019 caucvgprprlemnjltk 7020 caucvgprprlemnbj 7022 caucvgprprlemml 7023 caucvgprprlemlol 7027 caucvgprprlemupu 7029 gt0srpr 7064 lttrsr 7078 ltposr 7079 archsr 7097 |
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