Theorem frinfm 35025

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

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem frinfm

Step	Hyp	Ref	Expression
1		fri 5517	. . . . 5 ⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
2	1	ancom1s 651	. . . 4 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
3	2	exp43 439	. . 3 ⊢ (𝑅 Fr 𝐴 → (𝐵 ∈ 𝐶 → (𝐵 ⊆ 𝐴 → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))))
4	3	3imp2 1345	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
5		ssel2 3962	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
6	5	adantrr 715	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) → 𝑥 ∈ 𝐴)
7		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
8		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 5753	. . . . . . . . . . 11 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10	9	biimpi 218	. . . . . . . . . 10 ⊢ (𝑥◡𝑅𝑦 → 𝑦𝑅𝑥)
11	10	con3i 157	. . . . . . . . 9 ⊢ (¬ 𝑦𝑅𝑥 → ¬ 𝑥◡𝑅𝑦)
12	11	ralimi 3160	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 → ∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦)
13	12	ad2antll 727	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) → ∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦)
14		breq2 5070	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑦◡𝑅𝑧 ↔ 𝑦◡𝑅𝑥))
15	14	rspcev 3623	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥) → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)
16	15	ex 415	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
17	16	ralrimivw 3183	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
18	17	ad2antrl 726	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)) → ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
19	6, 13, 18	jca32 518	. . . . . 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 415	. . . . 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 3271	. . . 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 484	. . 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 1185	. 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 15	1 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  ∅c0 4291   class class class wbr 5066   Fr wfr 5511  ^◡ccnv 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-fr 5514  df-cnv 5563

This theorem is referenced by:  welb  35026