ltexprlem5 - Metamath Proof Explorer

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Theorem ltexprlem5 10993

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 10989	. . . 4 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅))
3		0pss 4410	. . . 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 10990	. . 3 ⊢ (𝐵 ∈ P → 𝐶 ⊊ Q)
7	6	adantr 480	. 2 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ⊊ Q)
8	1	ltexprlem3 10991	. . . . 5 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∀𝑧(𝑧 <_Q 𝑥 → 𝑧 ∈ 𝐶)))
9	1	ltexprlem4 10992	. . . . . 6 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑥 <_Q 𝑧)))
10		df-rex 3054	. . . . . 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 3124	. . 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 10940	. 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 2707 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ⊊ wpss 3915 ∅c0 4296 class class class wbr 5107 (class class class)co 7387 Qcnq 10805 +_Q cplq 10808 <_Q cltq 10811 Pcnp 10812

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 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-omul 8439 df-er 8671 df-ni 10825 df-pli 10826 df-mi 10827 df-lti 10828 df-plpq 10861 df-mpq 10862 df-ltpq 10863 df-enq 10864 df-nq 10865 df-erq 10866 df-plq 10867 df-mq 10868 df-1nq 10869 df-ltnq 10871 df-np 10934

This theorem is referenced by: ltexprlem6 10994 ltexprlem7 10995 ltexpri 10996

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