Theorem weinfcl 33613

Description: Closure of the infimum in a well-order. (Contributed by Thierry Arnoux, 29-Oct-2026.)

Hypotheses

Ref	Expression
weinfcl.1	⊢ (𝜑 → 𝑅 We 𝐴)
weinfcl.2	⊢ (𝜑 → 𝐵 ∈ 𝑉)
weinfcl.3	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
weinfcl.4	⊢ (𝜑 → 𝐵 ≠ ∅)

Assertion

Ref	Expression
weinfcl	⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)

Proof of Theorem weinfcl

Dummy variables 𝑖 𝑥 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		weinfcl.1	. . . . . 6 ⊢ (𝜑 → 𝑅 We 𝐴)
2		weso 5647	. . . . . 6 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → 𝑅 Or 𝐴)
4	3	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) → 𝑅 Or 𝐴)
5		weinfcl.3	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
6	5	sselda 3913	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑖 ∈ 𝐴)
7	6	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) → 𝑖 ∈ 𝐴)
8		breq1 5109	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥𝑅𝑖 ↔ 𝑦𝑅𝑖))
9	8	notbid 321	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (¬ 𝑥𝑅𝑖 ↔ ¬ 𝑦𝑅𝑖))
10	9	cbvralvw 3244	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑖)
11	10	bilani 510	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) → ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑖)
12	11	r19.21bi 3258	. . . 4 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝑖)
13		breq1 5109	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑗𝑅𝑦 ↔ 𝑖𝑅𝑦))
14		simpllr 788	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) ∧ (𝑦 ∈ 𝐴 ∧ 𝑖𝑅𝑦)) → 𝑖 ∈ 𝐵)
15		simprr 785	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) ∧ (𝑦 ∈ 𝐴 ∧ 𝑖𝑅𝑦)) → 𝑖𝑅𝑦)
16	13, 14, 15	rspcedvdw 3583	. . . 4 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) ∧ (𝑦 ∈ 𝐴 ∧ 𝑖𝑅𝑦)) → ∃𝑗 ∈ 𝐵 𝑗𝑅𝑦)
17	4, 7, 12, 16	eqinfd 9508	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) → inf(𝐵, 𝐴, 𝑅) = 𝑖)
18		simplr 781	. . 3 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) → 𝑖 ∈ 𝐵)
19	17, 18	eqeltrd 2864	. 2 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
20		wefr 5646	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
21	1, 20	syl 18	. . 3 ⊢ (𝜑 → 𝑅 Fr 𝐴)
22		weinfcl.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
23		weinfcl.4	. . 3 ⊢ (𝜑 → 𝐵 ≠ ∅)
24	21, 5, 22, 23	frd 5613	. 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑖)
25	19, 24	r19.29a 3174	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106   Or wor 5563   Fr wfr 5606   We wwe 5608  infcinf 9463

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-cnv 5664  df-iota 6515  df-riota 7403  df-sup 9464  df-inf 9465

This theorem is used by:  edompidlem  33940