Theorem fin2solem 34872

Description: Lemma for fin2so 34873. (Contributed by Brendan Leahy, 29-Jun-2019.)

Assertion

Ref	Expression
fin2solem	⊢ ((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) → (𝑦𝑅𝑧 → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧}))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝑅

Proof of Theorem fin2solem

Step	Hyp	Ref	Expression
1		ancom 463	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥) ∧ 𝑤 ∈ 𝑥) ↔ (𝑤 ∈ 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)))
2		3anass 1091	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥) ↔ (𝑤 ∈ 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)))
3	1, 2	bitr4i 280	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥) ∧ 𝑤 ∈ 𝑥) ↔ (𝑤 ∈ 𝑥 ∧ 𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥))
4		sotr 5491	. . . . . . . . 9 ⊢ ((𝑅 Or 𝑥 ∧ (𝑤 ∈ 𝑥 ∧ 𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) → ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))
5	3, 4	sylan2b 595	. . . . . . . 8 ⊢ ((𝑅 Or 𝑥 ∧ ((𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥) ∧ 𝑤 ∈ 𝑥)) → ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))
6	5	anassrs 470	. . . . . . 7 ⊢ (((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) ∧ 𝑤 ∈ 𝑥) → ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))
7	6	ancomsd 468	. . . . . 6 ⊢ (((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) ∧ 𝑤 ∈ 𝑥) → ((𝑦𝑅𝑧 ∧ 𝑤𝑅𝑦) → 𝑤𝑅𝑧))
8	7	expdimp 455	. . . . 5 ⊢ ((((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) ∧ 𝑤 ∈ 𝑥) ∧ 𝑦𝑅𝑧) → (𝑤𝑅𝑦 → 𝑤𝑅𝑧))
9	8	an32s 650	. . . 4 ⊢ ((((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) ∧ 𝑦𝑅𝑧) ∧ 𝑤 ∈ 𝑥) → (𝑤𝑅𝑦 → 𝑤𝑅𝑧))
10	9	ss2rabdv 4051	. . 3 ⊢ (((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ⊆ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})
11		breq1 5061	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤𝑅𝑧 ↔ 𝑦𝑅𝑧))
12	11	elrab 3679	. . . . . . 7 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ↔ (𝑦 ∈ 𝑥 ∧ 𝑦𝑅𝑧))
13	12	biimpri 230	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦𝑅𝑧) → 𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})
14	13	adantll 712	. . . . 5 ⊢ (((𝑅 Or 𝑥 ∧ 𝑦 ∈ 𝑥) ∧ 𝑦𝑅𝑧) → 𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})
15		sonr 5490	. . . . . . 7 ⊢ ((𝑅 Or 𝑥 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦𝑅𝑦)
16		breq1 5061	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤𝑅𝑦 ↔ 𝑦𝑅𝑦))
17	16	elrab 3679	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ↔ (𝑦 ∈ 𝑥 ∧ 𝑦𝑅𝑦))
18	17	simprbi 499	. . . . . . 7 ⊢ (𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} → 𝑦𝑅𝑦)
19	15, 18	nsyl 142	. . . . . 6 ⊢ ((𝑅 Or 𝑥 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦})
20	19	adantr 483	. . . . 5 ⊢ (((𝑅 Or 𝑥 ∧ 𝑦 ∈ 𝑥) ∧ 𝑦𝑅𝑧) → ¬ 𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦})
21		nelne1 3113	. . . . . 6 ⊢ ((𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦}) → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ≠ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦})
22	21	necomd 3071	. . . . 5 ⊢ ((𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ∧ ¬ 𝑦 ∈ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦}) → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ≠ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})
23	14, 20, 22	syl2anc 586	. . . 4 ⊢ (((𝑅 Or 𝑥 ∧ 𝑦 ∈ 𝑥) ∧ 𝑦𝑅𝑧) → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ≠ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})
24	23	adantlrr 719	. . 3 ⊢ (((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ≠ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})
25		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
26	25	rabex 5227	. . . . 5 ⊢ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ∈ V
27	26	brrpss 7446	. . . 4 ⊢ ({𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ↔ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ⊊ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})
28		df-pss 3953	. . . 4 ⊢ ({𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ⊊ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ↔ ({𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ⊆ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ∧ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ≠ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧}))
29	27, 28	bitri 277	. . 3 ⊢ ({𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ↔ ({𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ⊆ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧} ∧ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} ≠ {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧}))
30	10, 24, 29	sylanbrc 585	. 2 ⊢ (((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) ∧ 𝑦𝑅𝑧) → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧})
31	30	ex 415	1 ⊢ ((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) → (𝑦𝑅𝑧 → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2110   ≠ wne 3016  {crab 3142   ⊆ wss 3935   ⊊ wpss 3936   class class class wbr 5058   Or wor 5467   [⊊] crpss 7442

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-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  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-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-po 5468  df-so 5469  df-xp 5555  df-rel 5556  df-rpss 7443

This theorem is referenced by:  fin2so  34873