ltexprlem5 - Metamath Proof Explorer

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Theorem ltexprlem5 11059

Description: Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
ltexprlem.1	⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵)}

**Assertion**
Ref	Expression
ltexprlem5	⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ∈ P)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝐶 Allowed substitution hint: 𝐶(𝑦)

Proof of Theorem ltexprlem5 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . . 5 ⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵)}
2	1	ltexprlem1 11055	. . . 4 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅))
3		0pss 4427	. . . 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 11056	. . 3 ⊢ (𝐵 ∈ P → 𝐶 ⊊ Q)
7	6	adantr 480	. 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ⊊ Q)
8	1	ltexprlem3 11057	. . . . 5 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∀𝑧(𝑧 <_Q 𝑥 → 𝑧 ∈ 𝐶)))
9	1	ltexprlem4 11058	. . . . . 6 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <_Q 𝑧)))
10		df-rex 3062	. . . . . 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 3132	. . 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 11006	. 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 2109 {cab 2714 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 ⊊ wpss 3932 ∅c0 4313 class class class wbr 5124 (class class class)co 7410 Qcnq 10871 +_Q cplq 10874 <_Q cltq 10877 Pcnp 10878

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-omul 8490 df-er 8724 df-ni 10891 df-pli 10892 df-mi 10893 df-lti 10894 df-plpq 10927 df-mpq 10928 df-ltpq 10929 df-enq 10930 df-nq 10931 df-erq 10932 df-plq 10933 df-mq 10934 df-1nq 10935 df-ltnq 10937 df-np 11000

This theorem is referenced by: ltexprlem6 11060 ltexprlem7 11061 ltexpri 11062

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