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Theorem difinfsn 7166

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 7161	. . . . 5 ⊢ (ω ⊔ 1_o) ≈ ω
2		simp2 1000	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ω ≼ 𝐴)
3		endomtr 6849	. . . . 5 ⊢ (((ω ⊔ 1_o) ≈ ω ∧ ω ≼ 𝐴) → (ω ⊔ 1_o) ≼ 𝐴)
4	1, 2, 3	sylancr 414	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → (ω ⊔ 1_o) ≼ 𝐴)
5		brdomi 6808	. . . 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 7127	. . . . . . . 8 ⊢ (inl ↾ ω):ω–1-1→(ω ⊔ 1_o)
8		f1co 5475	. . . . . . . 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 425	. . . . . . 7 ⊢ (𝑓:(ω ⊔ 1_o)–1-1→𝐴 → (𝑓 ∘ (inl ↾ ω)):ω–1-1→𝐴)
10	9	ad2antlr 489	. . . . . 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 5463	. . . . . . . . . . . 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 5417	. . . . . . . . . 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 3183	. . . . . . . . 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 534	. . . . . . . . . . . . . . . 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 5463	. . . . . . . . . . . . . . . . . . . 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 5632	. . . . . . . . . . . . . . . . . . 19 ⊢ (((inl ↾ ω):ω⟶(ω ⊔ 1_o) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘((inl ↾ ω)‘𝑛)))
21	18, 19, 20	sylancr 414	. . . . . . . . . . . . . . . . . 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 5583	. . . . . . . . . . . . . . . . . . 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 5562	. . . . . . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . . . . 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 2236	. . . . . . . . . . . . . . 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 542	. . . . . . . . . . . . . . . 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 7117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ω → (inl‘𝑛) ∈ (ω ⊔ 1_o))
29	28	ad2antlr 489	. . . . . . . . . . . . . . . 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 6498	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 1_o
31		djurcl 7118	. . . . . . . . . . . . . . . . . 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 5816	. . . . . . . . . . . . . . . 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 1247	. . . . . . . . . . . . . . 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 7144	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ω ∧ ∅ ∈ 1_o) → (inl‘𝑛) ≠ (inr‘∅))
39	37, 30, 38	sylancl 413	. . . . . . . . . . . . . . 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 2388	. . . . . . . . . . . . . 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 662	. . . . . . . . . . . . 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 2570	. . . . . . . . . . . 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 5408	. . . . . . . . . . . . 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 2203	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (𝑠 = 𝐵 ↔ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
45	44	notbid 668	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (¬ 𝑠 = 𝐵 ↔ ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
46	45	ralrn 5700	. . . . . . . . . . . . 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 2585	. . . . . . . . . 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 3639	. . . . . . . . . 10 ⊢ (𝑠 ∈ {𝐵} ↔ 𝑠 = 𝐵)
51	49, 50	sylnibr 678	. . . . . . . . 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 3167	. . . . . . . 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 3189	. . . . . 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 5470	. . . . . 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 411	. . . . 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 5463	. . . . . . 7 ⊢ ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}))
58		omex 4629	. . . . . . 7 ⊢ ω ∈ V
59		fex 5791	. . . . . . 7 ⊢ (((𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}) ∧ ω ∈ V) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
60	57, 58, 59	sylancl 413	. . . . . 6 ⊢ ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
61		f1eq1 5458	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ (inl ↾ ω)) → (𝑔:ω–1-1→(𝐴 ∖ {𝐵}) ↔ (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵})))
62	61	spcegv 2852	. . . . . 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 1002	. . . . . . 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 1004	. . . . . . 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 2374	. . . . . 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 2196	. . . . . 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 7165	. . . . 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 5788	. . . . . 6 ⊢ (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))) ∈ V
76		f1eq1 5458	. . . . . 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 2859	. . . . 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 5463	. . . . . . . . 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 5698	. . . . . . 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 2209	. . . . . . . 8 ⊢ ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝑓‘(inr‘∅)) = 𝐵))
85	84	dcbid 839	. . . . . . 7 ⊢ ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦 ↔ DECID (𝑓‘(inr‘∅)) = 𝐵))
86	85	rspc2gv 2880	. . . . . 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 837	. . . . 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 799	. . 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 1913	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
92		reldom 6804	. . . . . 6 ⊢ Rel ≼
93	92	brrelex2i 4707	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
94		difexg 4174	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝐵}) ∈ V)
95	93, 94	syl 14	. . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ∈ V)
96	95	3ad2ant2 1021	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ∈ V)
97		brdomg 6807	. . 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 709 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∃wex 1506 ∈ wcel 2167 ≠ wne 2367 ∀wral 2475 Vcvv 2763 ∖ cdif 3154 ⊆ wss 3157 ∅c0 3450 ifcif 3561 {csn 3622 class class class wbr 4033 ↦ cmpt 4094 ωcom 4626 ran crn 4664 ↾ cres 4665 ∘ ccom 4667 Fn wfn 5253 ⟶wf 5254 –1-1→wf1 5255 ‘cfv 5258 1_oc1o 6467 ≈ cen 6797 ≼ cdom 6798 ⊔ cdju 7103 inlcinl 7111 inrcinr 7112

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-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-1st 6198 df-2nd 6199 df-1o 6474 df-er 6592 df-en 6800 df-dom 6801 df-dju 7104 df-inl 7113 df-inr 7114 df-case 7150

This theorem is referenced by: difinfinf 7167

W3C validator