Theorem frinfm 38589

Description: A subset of a well-founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
frinfm	⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))

Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦

Allowed substitution hint:   𝐶(𝑧)

Proof of Theorem frinfm

Step	Hyp	Ref	Expression
1		fri 5605	. . . . 5 ⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
2	1	ancom1s 666	. . . 4 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
3	2	exp43 442	. . 3 ⊢ (𝑅 Fr 𝐴 → (𝐵 ∈ 𝐶 → (𝐵 ⊆ 𝐴 → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))))
4	3	3imp2 1368	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
5		ssel2 3925	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
6	5	adantrr 730	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) → 𝑥 ∈ 𝐴)
7		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
8		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 5856	. . . . . . . . . . 11 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10	9	biimpi 219	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 → 𝑦𝑅𝑥)
11	10	con3i 155	. . . . . . . . 9 ⊢ (¬ 𝑦𝑅𝑥 → ¬ 𝑥◡𝑅𝑦)
12	11	ralimi 3099	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 → ∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦)
13	12	ad2antll 742	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) → ∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦)
14		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑦◡𝑅𝑧 ↔ 𝑦◡𝑅𝑥))
15	14	rspcev 3576	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥) → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)
16	15	ex 418	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
17	16	ralrimivw 3158	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
18	17	ad2antrl 741	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) → ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
19	6, 13, 18	jca32 525	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) → (𝑥 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
20	19	ex 418	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) → (𝑥 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))))
21	20	reximdv2 3172	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
22	21	adantl 487	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
23	22	3ad2antr2 1208	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
24	4, 23	mpd 16	1 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3898  ∅c0 4278   class class class wbr 5102   Fr wfr 5597  ^◡ccnv 5646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-fr 5600  df-cnv 5655

This theorem is used by:  welb  38590