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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 6453 | . 2 ⊢ <P = {〈x, y〉 ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))} | |
2 | opabssxp 4357 | . 2 ⊢ {〈x, y〉 ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))} ⊆ (P × P) | |
3 | 1, 2 | eqsstri 2969 | 1 ⊢ <P ⊆ (P × P) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∈ wcel 1390 ∃wrex 2301 ⊆ wss 2911 {copab 3808 × cxp 4286 ‘cfv 4845 1st c1st 5707 2nd c2nd 5708 Qcnq 6264 Pcnp 6275 <P cltp 6279 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-in 2918 df-ss 2925 df-opab 3810 df-xp 4294 df-iltp 6453 |
This theorem is referenced by: ltprordil 6565 ltexprlemm 6574 ltexprlemopl 6575 ltexprlemlol 6576 ltexprlemopu 6577 ltexprlemupu 6578 ltexprlemdisj 6580 ltexprlemloc 6581 ltexprlemfl 6583 ltexprlemrl 6584 ltexprlemfu 6585 ltexprlemru 6586 ltexpri 6587 ltaprlem 6591 gt0srpr 6676 lttrsr 6690 ltposr 6691 archsr 6708 |
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