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Theorem difinfsn 7430

Description: An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)

**Assertion**
Ref	Expression
difinfsn	⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ω ≼ (𝐴 ∖ {𝐵}))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦

Proof of Theorem difinfsn Dummy variables 𝑎 𝑓 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omp1eom 7425	. . . . 5 ⊢ (ω ⊔ 1_o) ≈ ω
2		simp2 1029	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ω ≼ 𝐴)
3		endomtr 7067	. . . . 5 ⊢ (((ω ⊔ 1_o) ≈ ω ∧ ω ≼ 𝐴) → (ω ⊔ 1_o) ≼ 𝐴)
4	1, 2, 3	sylancr 418	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → (ω ⊔ 1_o) ≼ 𝐴)
5		brdomi 7023	. . . 4 ⊢ ((ω ⊔ 1_o) ≼ 𝐴 → ∃𝑓 𝑓:(ω ⊔ 1_o)–1-1→𝐴)
6	4, 5	syl 14	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑓 𝑓:(ω ⊔ 1_o)–1-1→𝐴)
7		inlresf1 7391	. . . . . . . 8 ⊢ (inl ↾ ω):ω–1-1→(ω ⊔ 1_o)
8		f1co 5605	. . . . . . . 8 ⊢ ((𝑓:(ω ⊔ 1_o)–1-1→𝐴 ∧ (inl ↾ ω):ω–1-1→(ω ⊔ 1_o)) → (𝑓 ∘ (inl ↾ ω)):ω–1-1→𝐴)
9	7, 8	mpan2 429	. . . . . . 7 ⊢ (𝑓:(ω ⊔ 1_o)–1-1→𝐴 → (𝑓 ∘ (inl ↾ ω)):ω–1-1→𝐴)
10	9	ad2antlr 493	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)):ω–1-1→𝐴)
11		f1f 5593	. . . . . . . . . . . 12 ⊢ ((𝑓 ∘ (inl ↾ ω)):ω–1-1→𝐴 → (𝑓 ∘ (inl ↾ ω)):ω⟶𝐴)
12	10, 11	syl 14	. . . . . . . . . . 11 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)):ω⟶𝐴)
13	12	frnd 5538	. . . . . . . . . 10 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ran (𝑓 ∘ (inl ↾ ω)) ⊆ 𝐴)
14	13	sselda 3248	. . . . . . . . 9 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → 𝑠 ∈ 𝐴)
15		simpllr 540	. . . . . . . . . . . . . . . 16 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (𝑓‘(inr‘∅)) = 𝐵)
16		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
17		f1f 5593	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((inl ↾ ω):ω–1-1→(ω ⊔ 1_o) → (inl ↾ ω):ω⟶(ω ⊔ 1_o))
18	7, 17	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (inl ↾ ω):ω⟶(ω ⊔ 1_o)
19		simpr 110	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
20		fvco3 5770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((inl ↾ ω):ω⟶(ω ⊔ 1_o) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘((inl ↾ ω)‘𝑛)))
21	18, 19, 20	sylancr 418	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘((inl ↾ ω)‘𝑛)))
22	19	fvresd 5715	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ((inl ↾ ω)‘𝑛) = (inl‘𝑛))
23	22	fveq2d 5694	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → (𝑓‘((inl ↾ ω)‘𝑛)) = (𝑓‘(inl‘𝑛)))
24	21, 23	eqtrd 2271	. . . . . . . . . . . . . . . . 17 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘(inl‘𝑛)))
25	24	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘(inl‘𝑛)))
26	15, 16, 25	3eqtr2rd 2278	. . . . . . . . . . . . . . 15 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (𝑓‘(inl‘𝑛)) = (𝑓‘(inr‘∅)))
27		simp-4r 548	. . . . . . . . . . . . . . . 16 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → 𝑓:(ω ⊔ 1_o)–1-1→𝐴)
28		djulcl 7381	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ω → (inl‘𝑛) ∈ (ω ⊔ 1_o))
29	28	ad2antlr 493	. . . . . . . . . . . . . . . 16 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (inl‘𝑛) ∈ (ω ⊔ 1_o))
30		0lt1o 6703	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 1_o
31		djurcl 7382	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 1_o → (inr‘∅) ∈ (ω ⊔ 1_o))
32	30, 31	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (inr‘∅) ∈ (ω ⊔ 1_o)
33	32	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (inr‘∅) ∈ (ω ⊔ 1_o))
34		f1veqaeq 5965	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(ω ⊔ 1_o)–1-1→𝐴 ∧ ((inl‘𝑛) ∈ (ω ⊔ 1_o) ∧ (inr‘∅) ∈ (ω ⊔ 1_o))) → ((𝑓‘(inl‘𝑛)) = (𝑓‘(inr‘∅)) → (inl‘𝑛) = (inr‘∅)))
35	27, 29, 33, 34	syl12anc 1276	. . . . . . . . . . . . . . 15 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → ((𝑓‘(inl‘𝑛)) = (𝑓‘(inr‘∅)) → (inl‘𝑛) = (inr‘∅)))
36	26, 35	mpd 13	. . . . . . . . . . . . . 14 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (inl‘𝑛) = (inr‘∅))
37	19	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → 𝑛 ∈ ω)
38		djune 7408	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ω ∧ ∅ ∈ 1_o) → (inl‘𝑛) ≠ (inr‘∅))
39	37, 30, 38	sylancl 417	. . . . . . . . . . . . . . 15 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (inl‘𝑛) ≠ (inr‘∅))
40	39	neneqd 2441	. . . . . . . . . . . . . 14 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → ¬ (inl‘𝑛) = (inr‘∅))
41	36, 40	pm2.65da 671	. . . . . . . . . . . . 13 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
42	41	ralrimiva 2623	. . . . . . . . . . . 12 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑛 ∈ ω ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
43	12	ffnd 5529	. . . . . . . . . . . . 13 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)) Fn ω)
44		eqeq1 2245	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (𝑠 = 𝐵 ↔ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
45	44	notbid 677	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (¬ 𝑠 = 𝐵 ↔ ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
46	45	ralrn 5837	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∘ (inl ↾ ω)) Fn ω → (∀𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) ¬ 𝑠 = 𝐵 ↔ ∀𝑛 ∈ ω ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
47	43, 46	syl 14	. . . . . . . . . . . 12 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (∀𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) ¬ 𝑠 = 𝐵 ↔ ∀𝑛 ∈ ω ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
48	42, 47	mpbird 167	. . . . . . . . . . 11 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) ¬ 𝑠 = 𝐵)
49	48	r19.21bi 2638	. . . . . . . . . 10 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → ¬ 𝑠 = 𝐵)
50		velsn 3722	. . . . . . . . . 10 ⊢ (𝑠 ∈ {𝐵} ↔ 𝑠 = 𝐵)
51	49, 50	sylnibr 688	. . . . . . . . 9 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → ¬ 𝑠 ∈ {𝐵})
52	14, 51	eldifd 3230	. . . . . . . 8 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → 𝑠 ∈ (𝐴 ∖ {𝐵}))
53	52	ex 115	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) → 𝑠 ∈ (𝐴 ∖ {𝐵})))
54	53	ssrdv 3254	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ran (𝑓 ∘ (inl ↾ ω)) ⊆ (𝐴 ∖ {𝐵}))
55		f1ssr 5600	. . . . . 6 ⊢ (((𝑓 ∘ (inl ↾ ω)):ω–1-1→𝐴 ∧ ran (𝑓 ∘ (inl ↾ ω)) ⊆ (𝐴 ∖ {𝐵})) → (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}))
56	10, 54, 55	syl2anc 415	. . . . 5 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}))
57		f1f 5593	. . . . . . 7 ⊢ ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}))
58		omex 4735	. . . . . . 7 ⊢ ω ∈ V
59		fex 5937	. . . . . . 7 ⊢ (((𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}) ∧ ω ∈ V) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
60	57, 58, 59	sylancl 417	. . . . . 6 ⊢ ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
61		f1eq1 5588	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ (inl ↾ ω)) → (𝑔:ω–1-1→(𝐴 ∖ {𝐵}) ↔ (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵})))
62	61	spcegv 2913	. . . . . 6 ⊢ ((𝑓 ∘ (inl ↾ ω)) ∈ V → ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵})))
63	60, 62	mpcom 36	. . . . 5 ⊢ ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
64	56, 63	syl 14	. . . 4 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
65		simpl1 1031	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
66	65	adantr 276	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
67		simpl3 1033	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → 𝐵 ∈ 𝐴)
68	67	adantr 276	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → 𝐵 ∈ 𝐴)
69		simpr 110	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → 𝑓:(ω ⊔ 1_o)–1-1→𝐴)
70	69	adantr 276	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → 𝑓:(ω ⊔ 1_o)–1-1→𝐴)
71		simpr 110	. . . . . . 7 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ¬ (𝑓‘(inr‘∅)) = 𝐵)
72	71	neqned 2427	. . . . . 6 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓‘(inr‘∅)) ≠ 𝐵)
73		eqid 2238	. . . . . 6 ⊢ (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))) = (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎))))
74	66, 68, 70, 72, 73	difinfsnlem 7429	. . . . 5 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))):ω–1-1→(𝐴 ∖ {𝐵}))
75	58	mptex 5934	. . . . . 6 ⊢ (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))) ∈ V
76		f1eq1 5588	. . . . . 6 ⊢ (𝑔 = (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))) → (𝑔:ω–1-1→(𝐴 ∖ {𝐵}) ↔ (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))):ω–1-1→(𝐴 ∖ {𝐵})))
77	75, 76	spcev 2920	. . . . 5 ⊢ ((𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))):ω–1-1→(𝐴 ∖ {𝐵}) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
78	74, 77	syl 14	. . . 4 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
79		f1f 5593	. . . . . . . . 9 ⊢ (𝑓:(ω ⊔ 1_o)–1-1→𝐴 → 𝑓:(ω ⊔ 1_o)⟶𝐴)
80	69, 79	syl 14	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → 𝑓:(ω ⊔ 1_o)⟶𝐴)
81	32	a1i 9	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → (inr‘∅) ∈ (ω ⊔ 1_o))
82	80, 81	ffvelcdmd 5835	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → (𝑓‘(inr‘∅)) ∈ 𝐴)
83	82, 67	jca 306	. . . . . 6 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → ((𝑓‘(inr‘∅)) ∈ 𝐴 ∧ 𝐵 ∈ 𝐴))
84		eqeq12 2251	. . . . . . . 8 ⊢ ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝑓‘(inr‘∅)) = 𝐵))
85	84	dcbid 850	. . . . . . 7 ⊢ ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦 ↔ DECID (𝑓‘(inr‘∅)) = 𝐵))
86	85	rspc2gv 2942	. . . . . 6 ⊢ (((𝑓‘(inr‘∅)) ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID (𝑓‘(inr‘∅)) = 𝐵))
87	83, 65, 86	sylc 62	. . . . 5 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → DECID (𝑓‘(inr‘∅)) = 𝐵)
88		exmiddc 848	. . . . 5 ⊢ (DECID (𝑓‘(inr‘∅)) = 𝐵 → ((𝑓‘(inr‘∅)) = 𝐵 ∨ ¬ (𝑓‘(inr‘∅)) = 𝐵))
89	87, 88	syl 14	. . . 4 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → ((𝑓‘(inr‘∅)) = 𝐵 ∨ ¬ (𝑓‘(inr‘∅)) = 𝐵))
90	64, 78, 89	mpjaodan 810	. . 3 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:(ω ⊔ 1_o)–1-1→𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
91	6, 90	exlimddv 1954	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
92		reldom 7017	. . . . . 6 ⊢ Rel ≼
93	92	brrelex2i 4814	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
94		difexg 4270	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝐵}) ∈ V)
95	93, 94	syl 14	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ∈ V)
96	95	3ad2ant2 1050	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ∈ V)
97		brdomg 7022	. . 3 ⊢ ((𝐴 ∖ {𝐵}) ∈ V → (ω ≼ (𝐴 ∖ {𝐵}) ↔ ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵})))
98	96, 97	syl 14	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → (ω ≼ (𝐴 ∖ {𝐵}) ↔ ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵})))
99	91, 98	mpbird 167	1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ω ≼ (𝐴 ∖ {𝐵}))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 DECID wdc 846 ∧ w3a 1009 = wceq 1402 ∃wex 1545 ∈ wcel 2209 ≠ wne 2420 ∀wral 2528 Vcvv 2821 ∖ cdif 3217 ⊆ wss 3220 ∅c0 3520 ifcif 3635 {csn 3705 class class class wbr 4125 ↦ cmpt 4187 ωcom 4732 ran crn 4770 ↾ cres 4771 ∘ ccom 4773 Fn wfn 5367 ⟶wf 5368 –1-1→wf1 5369 ‘cfv 5372 1_oc1o 6670 ≈ cen 7010 ≼ cdom 7011 ⊔ cdju 7367 inlcinl 7375 inrcinr 7376

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730

This theorem depends on definitions: df-bi 117 df-dc 847 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-iord 4506 df-on 4508 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-1st 6364 df-2nd 6365 df-1o 6677 df-er 6797 df-en 7013 df-dom 7014 df-dju 7368 df-inl 7377 df-inr 7378 df-case 7414

This theorem is referenced by: difinfinf 7431

W3C validator