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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 7271 | . 2 ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} | |
2 | opabssxp 4608 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 3124 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∈ wcel 1480 ∃wrex 2415 ⊆ wss 3066 {copab 3983 × cxp 4532 ‘cfv 5118 1st c1st 6029 2nd c2nd 6030 Qcnq 7081 Pcnp 7092 <P cltp 7096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-in 3072 df-ss 3079 df-opab 3985 df-xp 4540 df-iltp 7271 |
This theorem is referenced by: ltprordil 7390 ltexprlemm 7401 ltexprlemopl 7402 ltexprlemlol 7403 ltexprlemopu 7404 ltexprlemupu 7405 ltexprlemdisj 7407 ltexprlemloc 7408 ltexprlemfl 7410 ltexprlemrl 7411 ltexprlemfu 7412 ltexprlemru 7413 ltexpri 7414 lteupri 7418 ltaprlem 7419 prplnqu 7421 caucvgprprlemk 7484 caucvgprprlemnkltj 7490 caucvgprprlemnkeqj 7491 caucvgprprlemnjltk 7492 caucvgprprlemnbj 7494 caucvgprprlemml 7495 caucvgprprlemlol 7499 caucvgprprlemupu 7501 suplocexprlemss 7516 suplocexprlemlub 7525 gt0srpr 7549 lttrsr 7563 ltposr 7564 archsr 7583 |
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