Theorem frirrg 4385

Description: A well-founded relation is irreflexive. This is the case where 𝐴 exists. (Contributed by Jim Kingdon, 21-Sep-2021.)

Assertion

Ref	Expression
frirrg	⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem frirrg

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

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐴 ⊆ (𝐴 ∖ {𝐵})) → 𝐴 ⊆ (𝐴 ∖ {𝐵}))
2		simpl3 1004	. . . 4 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐴 ⊆ (𝐴 ∖ {𝐵})) → 𝐵 ∈ 𝐴)
3	1, 2	sseldd 3184	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐴 ⊆ (𝐴 ∖ {𝐵})) → 𝐵 ∈ (𝐴 ∖ {𝐵}))
4		neldifsnd 3753	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐴 ⊆ (𝐴 ∖ {𝐵})) → ¬ 𝐵 ∈ (𝐴 ∖ {𝐵}))
5	3, 4	pm2.65da 662	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ⊆ (𝐴 ∖ {𝐵}))
6		simplr 528	. . . . . 6 ⊢ (((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) → 𝑥 ∈ 𝐴)
7		simplr 528	. . . . . . . . . . 11 ⊢ ((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐵𝑅𝐵)
8	7	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → 𝐵𝑅𝐵)
9		simpr 110	. . . . . . . . . 10 ⊢ ((((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
10	8, 9	breqtrrd 4061	. . . . . . . . 9 ⊢ ((((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → 𝐵𝑅𝑥)
11		breq1 4036	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝑥))
12		eleq1 2259	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ (𝐴 ∖ {𝐵}) ↔ 𝐵 ∈ (𝐴 ∖ {𝐵})))
13	11, 12	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) ↔ (𝐵𝑅𝑥 → 𝐵 ∈ (𝐴 ∖ {𝐵}))))
14		simplr 528	. . . . . . . . . 10 ⊢ ((((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})))
15		simpll3 1040	. . . . . . . . . . 11 ⊢ ((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴)
16	15	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴)
17	13, 14, 16	rspcdva 2873	. . . . . . . . 9 ⊢ ((((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝐵𝑅𝑥 → 𝐵 ∈ (𝐴 ∖ {𝐵})))
18	10, 17	mpd 13	. . . . . . . 8 ⊢ ((((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → 𝐵 ∈ (𝐴 ∖ {𝐵}))
19		neldifsnd 3753	. . . . . . . 8 ⊢ ((((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → ¬ 𝐵 ∈ (𝐴 ∖ {𝐵}))
20	18, 19	pm2.65da 662	. . . . . . 7 ⊢ (((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) → ¬ 𝑥 = 𝐵)
21		velsn 3639	. . . . . . 7 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
22	20, 21	sylnibr 678	. . . . . 6 ⊢ (((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) → ¬ 𝑥 ∈ {𝐵})
23	6, 22	eldifd 3167	. . . . 5 ⊢ (((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))) → 𝑥 ∈ (𝐴 ∖ {𝐵}))
24	23	ex 115	. . . 4 ⊢ ((((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵})))
25	24	ralrimiva 2570	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵})))
26		df-frind 4367	. . . . . . . 8 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
27		df-frfor 4366	. . . . . . . . 9 ⊢ ( FrFor 𝑅𝐴𝑠 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠))
28	27	albii 1484	. . . . . . . 8 ⊢ (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠(∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠))
29	26, 28	bitri 184	. . . . . . 7 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠(∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠))
30	29	biimpi 120	. . . . . 6 ⊢ (𝑅 Fr 𝐴 → ∀𝑠(∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠))
31	30	3ad2ant1 1020	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ∀𝑠(∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠))
32		difexg 4174	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {𝐵}) ∈ V)
33		eleq2 2260	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝐴 ∖ {𝐵}) → (𝑦 ∈ 𝑠 ↔ 𝑦 ∈ (𝐴 ∖ {𝐵})))
34	33	imbi2d 230	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝐴 ∖ {𝐵}) → ((𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) ↔ (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))))
35	34	ralbidv 2497	. . . . . . . . . . 11 ⊢ (𝑠 = (𝐴 ∖ {𝐵}) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵}))))
36		eleq2 2260	. . . . . . . . . . 11 ⊢ (𝑠 = (𝐴 ∖ {𝐵}) → (𝑥 ∈ 𝑠 ↔ 𝑥 ∈ (𝐴 ∖ {𝐵})))
37	35, 36	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑠 = (𝐴 ∖ {𝐵}) → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵}))))
38	37	ralbidv 2497	. . . . . . . . 9 ⊢ (𝑠 = (𝐴 ∖ {𝐵}) → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵}))))
39		sseq2 3207	. . . . . . . . 9 ⊢ (𝑠 = (𝐴 ∖ {𝐵}) → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ (𝐴 ∖ {𝐵})))
40	38, 39	imbi12d 234	. . . . . . . 8 ⊢ (𝑠 = (𝐴 ∖ {𝐵}) → ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐴 ⊆ (𝐴 ∖ {𝐵}))))
41	40	spcgv 2851	. . . . . . 7 ⊢ ((𝐴 ∖ {𝐵}) ∈ V → (∀𝑠(∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠) → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐴 ⊆ (𝐴 ∖ {𝐵}))))
42	32, 41	syl 14	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∀𝑠(∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠) → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐴 ⊆ (𝐴 ∖ {𝐵}))))
43	42	3ad2ant2 1021	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → (∀𝑠(∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠) → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐴 ⊆ (𝐴 ∖ {𝐵}))))
44	31, 43	mpd 13	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐴 ⊆ (𝐴 ∖ {𝐵})))
45	44	adantr 276	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐴 ⊆ (𝐴 ∖ {𝐵})))
46	25, 45	mpd 13	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵𝑅𝐵) → 𝐴 ⊆ (𝐴 ∖ {𝐵}))
47	5, 46	mtand 666	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 980  ∀wal 1362   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ∖ cdif 3154   ⊆ wss 3157  {csn 3622   class class class wbr 4033   FrFor wfrfor 4362   Fr wfr 4363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4151

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-frfor 4366  df-frind 4367

This theorem is referenced by:  efrirr  4388  wepo  4394  wetriext  4613