Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ltsopr | Structured version Visualization version GIF version |
Description: Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltsopr | ⊢ <P Or P |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssirr 4079 | . . . 4 ⊢ ¬ 𝑥 ⊊ 𝑥 | |
2 | ltprord 10454 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑥 ∈ P) → (𝑥<P 𝑥 ↔ 𝑥 ⊊ 𝑥)) | |
3 | 1, 2 | mtbiri 329 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑥 ∈ P) → ¬ 𝑥<P 𝑥) |
4 | 3 | anidms 569 | . 2 ⊢ (𝑥 ∈ P → ¬ 𝑥<P 𝑥) |
5 | psstr 4083 | . . 3 ⊢ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧) | |
6 | ltprord 10454 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥<P 𝑦 ↔ 𝑥 ⊊ 𝑦)) | |
7 | 6 | 3adant3 1128 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑦 ↔ 𝑥 ⊊ 𝑦)) |
8 | ltprord 10454 | . . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ 𝑦 ⊊ 𝑧)) | |
9 | 8 | 3adant1 1126 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ 𝑦 ⊊ 𝑧)) |
10 | 7, 9 | anbi12d 632 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → ((𝑥<P 𝑦 ∧ 𝑦<P 𝑧) ↔ (𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧))) |
11 | ltprord 10454 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑧 ↔ 𝑥 ⊊ 𝑧)) | |
12 | 11 | 3adant2 1127 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑧 ↔ 𝑥 ⊊ 𝑧)) |
13 | 10, 12 | imbi12d 347 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (((𝑥<P 𝑦 ∧ 𝑦<P 𝑧) → 𝑥<P 𝑧) ↔ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧))) |
14 | 5, 13 | mpbiri 260 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → ((𝑥<P 𝑦 ∧ 𝑦<P 𝑧) → 𝑥<P 𝑧)) |
15 | psslinpr 10455 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥)) | |
16 | biidd 264 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)) | |
17 | ltprord 10454 | . . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑥 ∈ P) → (𝑦<P 𝑥 ↔ 𝑦 ⊊ 𝑥)) | |
18 | 17 | ancoms 461 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑦<P 𝑥 ↔ 𝑦 ⊊ 𝑥)) |
19 | 6, 16, 18 | 3orbi123d 1431 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥<P 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦<P 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥))) |
20 | 15, 19 | mpbird 259 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥<P 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦<P 𝑥)) |
21 | 4, 14, 20 | issoi 5509 | 1 ⊢ <P Or P |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ w3o 1082 ∧ w3a 1083 ∈ wcel 2114 ⊊ wpss 3939 class class class wbr 5068 Or wor 5475 Pcnp 10283 <P cltp 10287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-omul 8109 df-er 8291 df-ni 10296 df-mi 10298 df-lti 10299 df-ltpq 10334 df-enq 10335 df-nq 10336 df-ltnq 10342 df-np 10405 df-ltp 10409 |
This theorem is referenced by: ltapr 10469 addcanpr 10470 suplem2pr 10477 ltsosr 10518 |
Copyright terms: Public domain | W3C validator |