Theorem weinflb 33614

Description: In a well-order, an infimum is a lower bound. (Contributed by Thierry Arnoux, 29-Oct-2026.)

Hypotheses

Ref	Expression
weinflb.1	⊢ (𝜑 → Rel 𝑅)
weinflb.2	⊢ (𝜑 → 𝑅 We 𝐴)
weinflb.3	⊢ (𝜑 → 𝐵 ∈ 𝑉)
weinflb.4	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
weinflb.5	⊢ (𝜑 → 𝐵 ≠ ∅)
weinflb.6	⊢ (𝜑 → 𝐶 ∈ 𝐵)

Assertion

Ref	Expression
weinflb	⊢ (𝜑 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))

Proof of Theorem weinflb

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

Step	Hyp	Ref	Expression
1		weinflb.6	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
2		weinflb.2	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐴)
3		weso 5647	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
5		weinflb.3	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
6		weinflb.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
7		weinflb.5	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅)
8		welb 6312	. . . . . 6 ⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → (◡𝑅 Or 𝐵 ∧ ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
9	2, 5, 6, 7, 8	syl13anc 1400	. . . . 5 ⊢ (𝜑 → (◡𝑅 Or 𝐵 ∧ ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
10	9	simprd 501	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
11	6	sselda 3913	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
12	11	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))) → 𝑥 ∈ 𝐴)
13		brcnvg 5865	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
14	13	notbid 321	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (¬ 𝑥◡𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
15	14	ralbidva 3187	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
16	15	biimpa 482	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦) → ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
17	16	adantll 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦) → ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
18		breq1 5109	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑦 ↔ 𝑥𝑅𝑦))
19		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐵)
20		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
21	18, 19, 20	rspcedvdw 3583	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)
22	21	ex 418	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))
23	22	ralrimiva 3158	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))
24	23	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))
25	17, 24	anim12dan 631	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
26	25	anasss 472	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))) → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
27	10, 12, 26	reximssdv 3184	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
28	4, 27	inflb 9512	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
29	1, 28	mpd 16	1 ⊢ (𝜑 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106   Or wor 5563   We wwe 5608  ^◡ccnv 5655  Rel wrel 5661  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-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  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