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Mirrors > Home > MPE Home > Th. List > ltexprlem5 | Structured version Visualization version GIF version |
Description: Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltexprlem.1 | ⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)} |
Ref | Expression |
---|---|
ltexprlem5 | ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltexprlem.1 | . . . . 5 ⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)} | |
2 | 1 | ltexprlem1 10452 | . . . 4 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅)) |
3 | 0pss 4395 | . . . 4 ⊢ (∅ ⊊ 𝐶 ↔ 𝐶 ≠ ∅) | |
4 | 2, 3 | syl6ibr 254 | . . 3 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → ∅ ⊊ 𝐶)) |
5 | 4 | imp 409 | . 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → ∅ ⊊ 𝐶) |
6 | 1 | ltexprlem2 10453 | . . 3 ⊢ (𝐵 ∈ P → 𝐶 ⊊ Q) |
7 | 6 | adantr 483 | . 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ⊊ Q) |
8 | 1 | ltexprlem3 10454 | . . . . 5 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶))) |
9 | 1 | ltexprlem4 10455 | . . . . . 6 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <Q 𝑧))) |
10 | df-rex 3144 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <Q 𝑧)) | |
11 | 9, 10 | syl6ibr 254 | . . . . 5 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧)) |
12 | 8, 11 | jcad 515 | . . . 4 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → (∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶) ∧ ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧))) |
13 | 12 | ralrimiv 3181 | . . 3 ⊢ (𝐵 ∈ P → ∀𝑥 ∈ 𝐶 (∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶) ∧ ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧)) |
14 | 13 | adantr 483 | . 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → ∀𝑥 ∈ 𝐶 (∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶) ∧ ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧)) |
15 | elnp 10403 | . 2 ⊢ (𝐶 ∈ P ↔ ((∅ ⊊ 𝐶 ∧ 𝐶 ⊊ Q) ∧ ∀𝑥 ∈ 𝐶 (∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶) ∧ ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧))) | |
16 | 5, 7, 14, 15 | syl21anbrc 1340 | 1 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ∈ P) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊊ wpss 3936 ∅c0 4290 class class class wbr 5058 (class class class)co 7150 Qcnq 10268 +Q cplq 10271 <Q cltq 10274 Pcnp 10275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-ltnq 10334 df-np 10397 |
This theorem is referenced by: ltexprlem6 10457 ltexprlem7 10458 ltexpri 10459 |
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