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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 7468 | . 2 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} | |
2 | opabssxp 4700 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 3187 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2148 ∃wrex 2456 ⊆ wss 3129 {copab 4063 × cxp 4624 ‘cfv 5216 1st c1st 6138 2nd c2nd 6139 Qcnq 7278 Pcnp 7289 <P cltp 7293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-in 3135 df-ss 3142 df-opab 4065 df-xp 4632 df-iltp 7468 |
This theorem is referenced by: ltprordil 7587 ltexprlemm 7598 ltexprlemopl 7599 ltexprlemlol 7600 ltexprlemopu 7601 ltexprlemupu 7602 ltexprlemdisj 7604 ltexprlemloc 7605 ltexprlemfl 7607 ltexprlemrl 7608 ltexprlemfu 7609 ltexprlemru 7610 ltexpri 7611 lteupri 7615 ltaprlem 7616 prplnqu 7618 caucvgprprlemk 7681 caucvgprprlemnkltj 7687 caucvgprprlemnkeqj 7688 caucvgprprlemnjltk 7689 caucvgprprlemnbj 7691 caucvgprprlemml 7692 caucvgprprlemlol 7696 caucvgprprlemupu 7698 suplocexprlemss 7713 suplocexprlemlub 7722 gt0srpr 7746 lttrsr 7760 ltposr 7761 archsr 7780 |
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