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Theorem difinfsn 7038
 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 7033 . . . . 5 (ω ⊔ 1o) ≈ ω
2 simp2 983 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ 𝐴)
3 endomtr 6732 . . . . 5 (((ω ⊔ 1o) ≈ ω ∧ ω ≼ 𝐴) → (ω ⊔ 1o) ≼ 𝐴)
41, 2, 3sylancr 411 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → (ω ⊔ 1o) ≼ 𝐴)
5 brdomi 6691 . . . 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 6999 . . . . . . . 8 (inl ↾ ω):ω–1-1→(ω ⊔ 1o)
8 f1co 5386 . . . . . . . 8 ((𝑓:(ω ⊔ 1o)–1-1𝐴 ∧ (inl ↾ ω):ω–1-1→(ω ⊔ 1o)) → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
97, 8mpan2 422 . . . . . . 7 (𝑓:(ω ⊔ 1o)–1-1𝐴 → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
109ad2antlr 481 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴)
11 f1f 5374 . . . . . . . . . . . 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 5328 . . . . . . . . . 10 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ran (𝑓 ∘ (inl ↾ ω)) ⊆ 𝐴)
1413sselda 3128 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → 𝑠𝐴)
15 simpllr 524 . . . . . . . . . . . . . . . 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 5374 . . . . . . . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . . . . . . 19 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
20 fvco3 5538 . . . . . . . . . . . . . . . . . . 19 (((inl ↾ ω):ω⟶(ω ⊔ 1o) ∧ 𝑛 ∈ ω) → ((𝑓 ∘ (inl ↾ ω))‘𝑛) = (𝑓‘((inl ↾ ω)‘𝑛)))
2118, 19, 20sylancr 411 . . . . . . . . . . . . . . . . . 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 5492 . . . . . . . . . . . . . . . . . . 19 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ((inl ↾ ω)‘𝑛) = (inl‘𝑛))
2322fveq2d 5471 . . . . . . . . . . . . . . . . . 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 2190 . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . 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 2197 . . . . . . . . . . . . . . 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 532 . . . . . . . . . . . . . . . 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 6989 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ω → (inl‘𝑛) ∈ (ω ⊔ 1o))
2928ad2antlr 481 . . . . . . . . . . . . . . . 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 6384 . . . . . . . . . . . . . . . . . 18 ∅ ∈ 1o
31 djurcl 6990 . . . . . . . . . . . . . . . . . 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 5716 . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . 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 7016 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ω ∧ ∅ ∈ 1o) → (inl‘𝑛) ≠ (inr‘∅))
3937, 30, 38sylancl 410 . . . . . . . . . . . . . . 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 2348 . . . . . . . . . . . . . 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 651 . . . . . . . . . . . . 13 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑛 ∈ ω) → ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
4241ralrimiva 2530 . . . . . . . . . . . 12 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑛 ∈ ω ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵)
4312ffnd 5319 . . . . . . . . . . . . 13 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓 ∘ (inl ↾ ω)) Fn ω)
44 eqeq1 2164 . . . . . . . . . . . . . . 15 (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (𝑠 = 𝐵 ↔ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
4544notbid 657 . . . . . . . . . . . . . 14 (𝑠 = ((𝑓 ∘ (inl ↾ ω))‘𝑛) → (¬ 𝑠 = 𝐵 ↔ ¬ ((𝑓 ∘ (inl ↾ ω))‘𝑛) = 𝐵))
4645ralrn 5604 . . . . . . . . . . . . 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 2545 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → ¬ 𝑠 = 𝐵)
50 velsn 3577 . . . . . . . . . 10 (𝑠 ∈ {𝐵} ↔ 𝑠 = 𝐵)
5149, 50sylnibr 667 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl ↾ ω))) → ¬ 𝑠 ∈ {𝐵})
5214, 51eldifd 3112 . . . . . . . 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 3134 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ (𝑓‘(inr‘∅)) = 𝐵) → ran (𝑓 ∘ (inl ↾ ω)) ⊆ (𝐴 ∖ {𝐵}))
55 f1ssr 5381 . . . . . 6 (((𝑓 ∘ (inl ↾ ω)):ω–1-1𝐴 ∧ ran (𝑓 ∘ (inl ↾ ω)) ⊆ (𝐴 ∖ {𝐵})) → (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}))
5610, 54, 55syl2anc 409 . . . . 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 5374 . . . . . . 7 ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}))
58 omex 4551 . . . . . . 7 ω ∈ V
59 fex 5693 . . . . . . 7 (((𝑓 ∘ (inl ↾ ω)):ω⟶(𝐴 ∖ {𝐵}) ∧ ω ∈ V) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
6057, 58, 59sylancl 410 . . . . . 6 ((𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵}) → (𝑓 ∘ (inl ↾ ω)) ∈ V)
61 f1eq1 5369 . . . . . . 7 (𝑔 = (𝑓 ∘ (inl ↾ ω)) → (𝑔:ω–1-1→(𝐴 ∖ {𝐵}) ↔ (𝑓 ∘ (inl ↾ ω)):ω–1-1→(𝐴 ∖ {𝐵})))
6261spcegv 2800 . . . . . 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 985 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
6665adantr 274 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
67 simpl3 987 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → 𝐵𝐴)
6867adantr 274 . . . . . 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 274 . . . . . 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 2334 . . . . . 6 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) ∧ ¬ (𝑓‘(inr‘∅)) = 𝐵) → (𝑓‘(inr‘∅)) ≠ 𝐵)
73 eqid 2157 . . . . . 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 7037 . . . . 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 5692 . . . . . 6 (𝑎 ∈ ω ↦ if((𝑓‘(inl‘𝑎)) = 𝐵, (𝑓‘(inr‘∅)), (𝑓‘(inl‘𝑎)))) ∈ V
76 f1eq1 5369 . . . . . 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 2807 . . . . 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 5374 . . . . . . . . 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 5602 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → (𝑓‘(inr‘∅)) ∈ 𝐴)
8382, 67jca 304 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ((𝑓‘(inr‘∅)) ∈ 𝐴𝐵𝐴))
84 eqeq12 2170 . . . . . . . 8 ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝑓‘(inr‘∅)) = 𝐵))
8584dcbid 824 . . . . . . 7 ((𝑥 = (𝑓‘(inr‘∅)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦DECID (𝑓‘(inr‘∅)) = 𝐵))
8685rspc2gv 2828 . . . . . 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 822 . . . . 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 788 . . 3 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:(ω ⊔ 1o)–1-1𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
916, 90exlimddv 1878 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ∃𝑔 𝑔:ω–1-1→(𝐴 ∖ {𝐵}))
92 reldom 6687 . . . . . 6 Rel ≼
9392brrelex2i 4629 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94 difexg 4105 . . . . 5 (𝐴 ∈ V → (𝐴 ∖ {𝐵}) ∈ V)
9593, 94syl 14 . . . 4 (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ∈ V)
96953ad2ant2 1004 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ∈ V)
97 brdomg 6690 . . 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 698  DECID wdc 820   ∧ w3a 963   = wceq 1335  ∃wex 1472   ∈ wcel 2128   ≠ wne 2327  ∀wral 2435  Vcvv 2712   ∖ cdif 3099   ⊆ wss 3102  ∅c0 3394  ifcif 3505  {csn 3560   class class class wbr 3965   ↦ cmpt 4025  ωcom 4548  ran crn 4586   ↾ cres 4587   ∘ ccom 4589   Fn wfn 5164  ⟶wf 5165  –1-1→wf1 5166  ‘cfv 5169  1oc1o 6353   ≈ cen 6680   ≼ cdom 6681   ⊔ cdju 6975  inlcinl 6983  inrcinr 6984 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-1st 6085  df-2nd 6086  df-1o 6360  df-er 6477  df-en 6683  df-dom 6684  df-dju 6976  df-inl 6985  df-inr 6986  df-case 7022 This theorem is referenced by:  difinfinf  7039
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