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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 11076 | . . . 4 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅)) |
| 3 | 0pss 4447 | . . . 4 ⊢ (∅ ⊊ 𝐶 ↔ 𝐶 ≠ ∅) | |
| 4 | 2, 3 | imbitrrdi 252 | . . 3 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → ∅ ⊊ 𝐶)) |
| 5 | 4 | imp 406 | . 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → ∅ ⊊ 𝐶) |
| 6 | 1 | ltexprlem2 11077 | . . 3 ⊢ (𝐵 ∈ P → 𝐶 ⊊ Q) |
| 7 | 6 | adantr 480 | . 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ⊊ Q) |
| 8 | 1 | ltexprlem3 11078 | . . . . 5 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶))) |
| 9 | 1 | ltexprlem4 11079 | . . . . . 6 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <Q 𝑧))) |
| 10 | df-rex 3071 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <Q 𝑧)) | |
| 11 | 9, 10 | imbitrrdi 252 | . . . . 5 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧)) |
| 12 | 8, 11 | jcad 512 | . . . 4 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → (∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶) ∧ ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧))) |
| 13 | 12 | ralrimiv 3145 | . . 3 ⊢ (𝐵 ∈ P → ∀𝑥 ∈ 𝐶 (∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶) ∧ ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧)) |
| 14 | 13 | adantr 480 | . 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → ∀𝑥 ∈ 𝐶 (∀𝑧(𝑧 <Q 𝑥 → 𝑧 ∈ 𝐶) ∧ ∃𝑧 ∈ 𝐶 𝑥 <Q 𝑧)) |
| 15 | elnp 11027 | . 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 1345 | 1 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ∈ P) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊊ wpss 3952 ∅c0 4333 class class class wbr 5143 (class class class)co 7431 Qcnq 10892 +Q cplq 10895 <Q cltq 10898 Pcnp 10899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ni 10912 df-pli 10913 df-mi 10914 df-lti 10915 df-plpq 10948 df-mpq 10949 df-ltpq 10950 df-enq 10951 df-nq 10952 df-erq 10953 df-plq 10954 df-mq 10955 df-1nq 10956 df-ltnq 10958 df-np 11021 |
| This theorem is referenced by: ltexprlem6 11081 ltexprlem7 11082 ltexpri 11083 |
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