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Theorem difinfsn 6937
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 6932 . . . . 5 (ω ⊔ 1o) ≈ ω
2 simp2 965 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
3 endomtr 6638 . . . . 5 (((ω ⊔ 1o) ≈ ω ∧ ω ≼ 𝐴) → (ω ⊔ 1o) ≼ 𝐴)
41, 2, 3sylancr 408 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → (ω ⊔ 1o) ≼ 𝐴)
5 brdomi 6597 . . . 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 6898 . . . . . . . 8 (inl ↾ ω):ω–1-1→(ω ⊔ 1o)
8 f1co 5298 . . . . . . . 8 ((𝑓:(ω ⊔ 1o)–1-1𝐴 ∧ (inl ↾ ω):ω–1-1→(ω ⊔ 1o)) → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
97, 8mpan2 419 . . . . . . 7 (𝑓:(ω ⊔ 1o)–1-1𝐴 → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
109ad2antlr 478 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
11 f1f 5286 . . . . . . . . . . . 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 5240 . . . . . . . . . 10 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ran (𝑓 ∘ (inl ↾ ω)) ⊆ 𝐴)
1413sselda 3063 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → 𝑠𝐴)
15 simpllr 506 . . . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . . . 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 5286 . . . . . . . . . . . . . . . . . . . 20 ((inl ↾ ω):ω–1-1→(ω ⊔ 1o) → (inl ↾ ω):ω⟶(ω ⊔ 1o))
187, 17ax-mp 7 . . . . . . . . . . . . . . . . . . 19 (inl ↾ ω):ω⟶(ω ⊔ 1o)
19 simpr 109 . . . . . . . . . . . . . . . . . . 19 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
20 fvco3 5446 . . . . . . . . . . . . . . . . . . 19 (((inl ↾ ω):ω⟶(ω ⊔ 1o) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘((inl ↾ ω)‘𝑛)))
2118, 19, 20sylancr 408 . . . . . . . . . . . . . . . . . 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 5400 . . . . . . . . . . . . . . . . . . 19 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ((inl ↾ ω)‘𝑛) = (inl‘𝑛))
2322fveq2d 5379 . . . . . . . . . . . . . . . . . 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 2147 . . . . . . . . . . . . . . . . 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 272 . . . . . . . . . . . . . . . 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 2154 . . . . . . . . . . . . . . 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 514 . . . . . . . . . . . . . . . 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 6888 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ω → (inl‘𝑛) ∈ (ω ⊔ 1o))
2928ad2antlr 478 . . . . . . . . . . . . . . . 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 6291 . . . . . . . . . . . . . . . . . 18 ∅ ∈ 1o
31 djurcl 6889 . . . . . . . . . . . . . . . . . 18 (∅ ∈ 1o → (inr‘∅) ∈ (ω ⊔ 1o))
3230, 31ax-mp 7 . . . . . . . . . . . . . . . . 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 5624 . . . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . . 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 272 . . . . . . . . . . . . . . . 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 6915 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ω ∧ ∅ ∈ 1o) → (inl‘𝑛) ≠ (inr‘∅))
3937, 30, 38sylancl 407 . . . . . . . . . . . . . . 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 2303 . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . 13 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
4241ralrimiva 2479 . . . . . . . . . . . 12 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑛 ∈ ω ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
4312ffnd 5231 . . . . . . . . . . . . 13 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)) Fn ω)
44 eqeq1 2121 . . . . . . . . . . . . . . 15 (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (𝑠 = 𝐵 ↔ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
4544notbid 639 . . . . . . . . . . . . . 14 (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (¬ 𝑠 = 𝐵 ↔ ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
4645ralrn 5512 . . . . . . . . . . . . 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 166 . . . . . . . . . . 11 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) ¬ 𝑠 = 𝐵)
4948r19.21bi 2494 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → ¬ 𝑠 = 𝐵)
50 velsn 3510 . . . . . . . . . 10 (𝑠 ∈ {𝐵} ↔ 𝑠 = 𝐵)
5149, 50sylnibr 649 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → ¬ 𝑠 ∈ {𝐵})
5214, 51eldifd 3047 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → 𝑠 ∈ (𝐴 ∖ {𝐵}))
5352ex 114 . . . . . . 7 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω)) → 𝑠 ∈ (𝐴 ∖ {𝐵})))
5453ssrdv 3069 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ran (𝑓 ∘ (inl ↾ ω)) ⊆ (𝐴 ∖ {𝐵}))
55 f1ssr 5293 . . . . . 6 (((𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴 ∧ ran (𝑓 ∘ (inl ↾ ω)) ⊆ (𝐴 ∖ {𝐵})) → (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}))
5610, 54, 55syl2anc 406 . . . . 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 5286 . . . . . . 7 ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}))
58 omex 4467 . . . . . . 7 ω ∈ V
59 fex 5601 . . . . . . 7 (((𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}) ∧ ω ∈ V) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
6057, 58, 59sylancl 407 . . . . . 6 ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
61 f1eq1 5281 . . . . . . 7 (𝑔 = (𝑓 ∘ (inl ↾ ω)) → (𝑔:ω–1-1→(𝐴 ∖ {𝐵}) ↔ (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵})))
6261spcegv 2745 . . . . . 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 967 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
6665adantr 272 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
67 simpl3 969 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → 𝐵𝐴)
6867adantr 272 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → 𝐵𝐴)
69 simpr 109 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → 𝑓:(ω ⊔ 1o)–1-1𝐴)
7069adantr 272 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → 𝑓:(ω ⊔ 1o)–1-1𝐴)
71 simpr 109 . . . . . . 7 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ¬ (𝑓‘(inr‘∅)) = 𝐵)
7271neqned 2289 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓‘(inr‘∅)) ≠ 𝐵)
73 eqid 2115 . . . . . 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 6936 . . . . 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 5600 . . . . . 6 (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))) ∈ V
76 f1eq1 5281 . . . . . 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 2751 . . . . 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 5286 . . . . . . . . 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, 81ffvelrnd 5510 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → (𝑓‘(inr‘∅)) ∈ 𝐴)
8382, 67jca 302 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ((𝑓‘(inr‘∅)) ∈ 𝐴𝐵𝐴))
84 eqeq12 2127 . . . . . . . 8 ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝑓‘(inr‘∅)) = 𝐵))
8584dcbid 806 . . . . . . 7 ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦DECID (𝑓‘(inr‘∅)) = 𝐵))
8685rspc2gv 2771 . . . . . 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 804 . . . . 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 770 . . 3 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
916, 90exlimddv 1852 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
92 reldom 6593 . . . . . 6 Rel ≼
9392brrelex2i 4543 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94 difexg 4029 . . . . 5 (𝐴 ∈ V → (𝐴 ∖ {𝐵}) ∈ V)
9593, 94syl 14 . . . 4 (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ∈ V)
96953ad2ant2 986 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ∈ V)
97 brdomg 6596 . . 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 166 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴 ∖ {𝐵}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  DECID wdc 802  w3a 945   = wceq 1314  wex 1451  wcel 1463  wne 2282  wral 2390  Vcvv 2657  cdif 3034  wss 3037  c0 3329  ifcif 3440  {csn 3493   class class class wbr 3895  cmpt 3949  ωcom 4464  ran crn 4500  cres 4501  ccom 4503   Fn wfn 5076  wf 5077  1-1wf1 5078  cfv 5081  1oc1o 6260  cen 6586  cdom 6587  cdju 6874  inlcinl 6882  inrcinr 6883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-1st 5992  df-2nd 5993  df-1o 6267  df-er 6383  df-en 6589  df-dom 6590  df-dju 6875  df-inl 6884  df-inr 6885  df-case 6921
This theorem is referenced by:  difinfinf  6938
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