ltexprlem5 - Metamath Proof Explorer

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Theorem ltexprlem5 11057

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 11053	. . . 4 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅))
3		0pss 4440	. . . 4 ⊢ (∅ ⊊ 𝐶 ↔ 𝐶 ≠ ∅)
4	2, 3	imbitrrdi 251	. . 3 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → ∅ ⊊ 𝐶))
5	4	imp 406	. 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → ∅ ⊊ 𝐶)
6	1	ltexprlem2 11054	. . 3 ⊢ (𝐵 ∈ P → 𝐶 ⊊ Q)
7	6	adantr 480	. 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ⊊ Q)
8	1	ltexprlem3 11055	. . . . 5 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∀𝑧(𝑧 <_Q 𝑥 → 𝑧 ∈ 𝐶)))
9	1	ltexprlem4 11056	. . . . . 6 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <_Q 𝑧)))
10		df-rex 3067	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 𝑥 <_Q 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <_Q 𝑧))
11	9, 10	imbitrrdi 251	. . . . 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 3141	. . 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 11004	. 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 1342	1 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ∈ P)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1532 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2705 ≠ wne 2936 ∀wral 3057 ∃wrex 3066 ⊊ wpss 3946 ∅c0 4318 class class class wbr 5142 (class class class)co 7414 Qcnq 10869 +_Q cplq 10872 <_Q cltq 10875 Pcnp 10876

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-omul 8485 df-er 8718 df-ni 10889 df-pli 10890 df-mi 10891 df-lti 10892 df-plpq 10925 df-mpq 10926 df-ltpq 10927 df-enq 10928 df-nq 10929 df-erq 10930 df-plq 10931 df-mq 10932 df-1nq 10933 df-ltnq 10935 df-np 10998

This theorem is referenced by: ltexprlem6 11058 ltexprlem7 11059 ltexpri 11060

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