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Theorem difinfsn 7093
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 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → 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.
StepHypRef Expression
1 omp1eom 7088 . . . . 5 (ω ⊔ 1o) ≈ ω
2 simp2 998 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
3 endomtr 6784 . . . . 5 (((ω ⊔ 1o) ≈ ω ∧ ω ≼ 𝐴) → (ω ⊔ 1o) ≼ 𝐴)
41, 2, 3sylancr 414 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → (ω ⊔ 1o) ≼ 𝐴)
5 brdomi 6743 . . . 4 ((ω ⊔ 1o) ≼ 𝐴 → ∃𝑓 𝑓:(ω ⊔ 1o)–1-1𝐴)
64, 5syl 14 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ∃𝑓 𝑓:(ω ⊔ 1o)–1-1𝐴)
7 inlresf1 7054 . . . . . . . 8 (inl ↾ ω):ω–1-1→(ω ⊔ 1o)
8 f1co 5429 . . . . . . . 8 ((𝑓:(ω ⊔ 1o)–1-1𝐴 ∧ (inl ↾ ω):ω–1-1→(ω ⊔ 1o)) → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
97, 8mpan2 425 . . . . . . 7 (𝑓:(ω ⊔ 1o)–1-1𝐴 → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
109ad2antlr 489 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
11 f1f 5417 . . . . . . . . . . . 12 ((𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴 → (𝑓 ∘ (inl ↾ ω)):ω⟶𝐴)
1210, 11syl 14 . . . . . . . . . . 11 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)):ω⟶𝐴)
1312frnd 5371 . . . . . . . . . 10 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ran (𝑓 ∘ (inl ↾ ω)) ⊆ 𝐴)
1413sselda 3155 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → 𝑠𝐴)
15 simpllr 534 . . . . . . . . . . . . . . . 16 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (𝑓‘(inr‘∅)) = 𝐵)
16 simpr 110 . . . . . . . . . . . . . . . 16 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
17 f1f 5417 . . . . . . . . . . . . . . . . . . . 20 ((inl ↾ ω):ω–1-1→(ω ⊔ 1o) → (inl ↾ ω):ω⟶(ω ⊔ 1o))
187, 17ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (inl ↾ ω):ω⟶(ω ⊔ 1o)
19 simpr 110 . . . . . . . . . . . . . . . . . . 19 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
20 fvco3 5583 . . . . . . . . . . . . . . . . . . 19 (((inl ↾ ω):ω⟶(ω ⊔ 1o) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘((inl ↾ ω)‘𝑛)))
2118, 19, 20sylancr 414 . . . . . . . . . . . . . . . . . 18 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘((inl ↾ ω)‘𝑛)))
2219fvresd 5536 . . . . . . . . . . . . . . . . . . 19 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ((inl ↾ ω)‘𝑛) = (inl‘𝑛))
2322fveq2d 5515 . . . . . . . . . . . . . . . . . 18 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → (𝑓‘((inl ↾ ω)‘𝑛)) = (𝑓‘(inl‘𝑛)))
2421, 23eqtrd 2210 . . . . . . . . . . . . . . . . 17 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘(inl‘𝑛)))
2524adantr 276 . . . . . . . . . . . . . . . 16 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘(inl‘𝑛)))
2615, 16, 253eqtr2rd 2217 . . . . . . . . . . . . . . 15 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (𝑓‘(inl‘𝑛)) = (𝑓‘(inr‘∅)))
27 simp-4r 542 . . . . . . . . . . . . . . . 16 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → 𝑓:(ω ⊔ 1o)–1-1𝐴)
28 djulcl 7044 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ω → (inl‘𝑛) ∈ (ω ⊔ 1o))
2928ad2antlr 489 . . . . . . . . . . . . . . . 16 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (inl‘𝑛) ∈ (ω ⊔ 1o))
30 0lt1o 6435 . . . . . . . . . . . . . . . . . 18 ∅ ∈ 1o
31 djurcl 7045 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 1o → (inr‘∅) ∈ (ω ⊔ 1o))
3230, 31ax-mp 5 . . . . . . . . . . . . . . . . 17 (inr‘∅) ∈ (ω ⊔ 1o)
3332a1i 9 . . . . . . . . . . . . . . . 16 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (inr‘∅) ∈ (ω ⊔ 1o))
34 f1veqaeq 5764 . . . . . . . . . . . . . . . 16 ((𝑓:(ω ⊔ 1o)–1-1𝐴 ∧ ((inl‘𝑛) ∈ (ω ⊔ 1o) ∧ (inr‘∅) ∈ (ω ⊔ 1o))) → ((𝑓‘(inl‘𝑛)) = (𝑓‘(inr‘∅)) → (inl‘𝑛) = (inr‘∅)))
3527, 29, 33, 34syl12anc 1236 . . . . . . . . . . . . . . 15 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → ((𝑓‘(inl‘𝑛)) = (𝑓‘(inr‘∅)) → (inl‘𝑛) = (inr‘∅)))
3626, 35mpd 13 . . . . . . . . . . . . . 14 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (inl‘𝑛) = (inr‘∅))
3719adantr 276 . . . . . . . . . . . . . . . 16 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → 𝑛 ∈ ω)
38 djune 7071 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ω ∧ ∅ ∈ 1o) → (inl‘𝑛) ≠ (inr‘∅))
3937, 30, 38sylancl 413 . . . . . . . . . . . . . . 15 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → (inl‘𝑛) ≠ (inr‘∅))
4039neneqd 2368 . . . . . . . . . . . . . 14 ((((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) ∧ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵) → ¬ (inl‘𝑛) = (inr‘∅))
4136, 40pm2.65da 661 . . . . . . . . . . . . 13 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
4241ralrimiva 2550 . . . . . . . . . . . 12 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑛 ∈ ω ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
4312ffnd 5362 . . . . . . . . . . . . 13 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)) Fn ω)
44 eqeq1 2184 . . . . . . . . . . . . . . 15 (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (𝑠 = 𝐵 ↔ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
4544notbid 667 . . . . . . . . . . . . . 14 (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (¬ 𝑠 = 𝐵 ↔ ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
4645ralrn 5650 . . . . . . . . . . . . 13 ((𝑓 ∘ (inl ↾ ω)) Fn ω → (∀𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) ¬ 𝑠 = 𝐵 ↔ ∀𝑛 ∈ ω ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
4743, 46syl 14 . . . . . . . . . . . 12 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (∀𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) ¬ 𝑠 = 𝐵 ↔ ∀𝑛 ∈ ω ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
4842, 47mpbird 167 . . . . . . . . . . 11 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) ¬ 𝑠 = 𝐵)
4948r19.21bi 2565 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → ¬ 𝑠 = 𝐵)
50 velsn 3608 . . . . . . . . . 10 (𝑠 ∈ {𝐵} ↔ 𝑠 = 𝐵)
5149, 50sylnibr 677 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → ¬ 𝑠 ∈ {𝐵})
5214, 51eldifd 3139 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → 𝑠 ∈ (𝐴 ∖ {𝐵}))
5352ex 115 . . . . . . 7 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) → 𝑠 ∈ (𝐴 ∖ {𝐵})))
5453ssrdv 3161 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ran (𝑓 ∘ (inl ↾ ω)) ⊆ (𝐴 ∖ {𝐵}))
55 f1ssr 5424 . . . . . 6 (((𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴 ∧ ran (𝑓 ∘ (inl ↾ ω)) ⊆ (𝐴 ∖ {𝐵})) → (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}))
5610, 54, 55syl2anc 411 . . . . 5 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}))
57 f1f 5417 . . . . . . 7 ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}))
58 omex 4589 . . . . . . 7 ω ∈ V
59 fex 5741 . . . . . . 7 (((𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}) ∧ ω ∈ V) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
6057, 58, 59sylancl 413 . . . . . 6 ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
61 f1eq1 5412 . . . . . . 7 (𝑔 = (𝑓 ∘ (inl ↾ ω)) → (𝑔:ω–1-1→(𝐴 ∖ {𝐵}) ↔ (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵})))
6261spcegv 2825 . . . . . 6 ((𝑓 ∘ (inl ↾ ω)) ∈ V → ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵})))
6360, 62mpcom 36 . . . . 5 ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
6456, 63syl 14 . . . 4 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
65 simpl1 1000 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
6665adantr 276 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
67 simpl3 1002 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → 𝐵𝐴)
6867adantr 276 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → 𝐵𝐴)
69 simpr 110 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → 𝑓:(ω ⊔ 1o)–1-1𝐴)
7069adantr 276 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → 𝑓:(ω ⊔ 1o)–1-1𝐴)
71 simpr 110 . . . . . . 7 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ¬ (𝑓‘(inr‘∅)) = 𝐵)
7271neqned 2354 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓‘(inr‘∅)) ≠ 𝐵)
73 eqid 2177 . . . . . 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))))
7466, 68, 70, 72, 73difinfsnlem 7092 . . . . 5 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))):ω–1-1→(𝐴 ∖ {𝐵}))
7558mptex 5738 . . . . . 6 (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))) ∈ V
76 f1eq1 5412 . . . . . 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})))
7775, 76spcev 2832 . . . . 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}))
7874, 77syl 14 . . . 4 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
79 f1f 5417 . . . . . . . . 9 (𝑓:(ω ⊔ 1o)–1-1𝐴𝑓:(ω ⊔ 1o)⟶𝐴)
8069, 79syl 14 . . . . . . . 8 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → 𝑓:(ω ⊔ 1o)⟶𝐴)
8132a1i 9 . . . . . . . 8 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → (inr‘∅) ∈ (ω ⊔ 1o))
8280, 81ffvelcdmd 5648 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → (𝑓‘(inr‘∅)) ∈ 𝐴)
8382, 67jca 306 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ((𝑓‘(inr‘∅)) ∈ 𝐴𝐵𝐴))
84 eqeq12 2190 . . . . . . . 8 ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝑓‘(inr‘∅)) = 𝐵))
8584dcbid 838 . . . . . . 7 ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦DECID (𝑓‘(inr‘∅)) = 𝐵))
8685rspc2gv 2853 . . . . . 6 (((𝑓‘(inr‘∅)) ∈ 𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦DECID (𝑓‘(inr‘∅)) = 𝐵))
8783, 65, 86sylc 62 . . . . 5 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → DECID (𝑓‘(inr‘∅)) = 𝐵)
88 exmiddc 836 . . . . 5 (DECID (𝑓‘(inr‘∅)) = 𝐵 → ((𝑓‘(inr‘∅)) = 𝐵 ∨ ¬ (𝑓‘(inr‘∅)) = 𝐵))
8987, 88syl 14 . . . 4 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ((𝑓‘(inr‘∅)) = 𝐵 ∨ ¬ (𝑓‘(inr‘∅)) = 𝐵))
9064, 78, 89mpjaodan 798 . . 3 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
916, 90exlimddv 1898 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
92 reldom 6739 . . . . . 6 Rel ≼
9392brrelex2i 4667 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94 difexg 4141 . . . . 5 (𝐴 ∈ V → (𝐴 ∖ {𝐵}) ∈ V)
9593, 94syl 14 . . . 4 (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ∈ V)
96953ad2ant2 1019 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ∈ V)
97 brdomg 6742 . . 3 ((𝐴 ∖ {𝐵}) ∈ V → (ω ≼ (𝐴 ∖ {𝐵}) ↔ ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵})))
9896, 97syl 14 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → (ω ≼ (𝐴 ∖ {𝐵}) ↔ ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵})))
9991, 98mpbird 167 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴 ∖ {𝐵}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wne 2347  wral 2455  Vcvv 2737  cdif 3126  wss 3129  c0 3422  ifcif 3534  {csn 3591   class class class wbr 4000  cmpt 4061  ωcom 4586  ran crn 4624  cres 4625  ccom 4627   Fn wfn 5207  wf 5208  1-1wf1 5209  cfv 5212  1oc1o 6404  cen 6732  cdom 6733  cdju 7030  inlcinl 7038  inrcinr 7039
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-er 6529  df-en 6735  df-dom 6736  df-dju 7031  df-inl 7040  df-inr 7041  df-case 7077
This theorem is referenced by:  difinfinf  7094
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