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Theorem cfsuc 9871
Description: Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
Assertion
Ref Expression
cfsuc (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)

Proof of Theorem cfsuc
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sucelon 7596 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2 cfval 9861 . . 3 (suc 𝐴 ∈ On → (cf‘suc 𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
31, 2sylbi 220 . 2 (𝐴 ∈ On → (cf‘suc 𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
4 cardsn 9585 . . . . . 6 (𝐴 ∈ On → (card‘{𝐴}) = 1o)
54eqcomd 2743 . . . . 5 (𝐴 ∈ On → 1o = (card‘{𝐴}))
6 snidg 4575 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7 elsuci 6279 . . . . . . . . 9 (𝑧 ∈ suc 𝐴 → (𝑧𝐴𝑧 = 𝐴))
8 onelss 6255 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧𝐴𝑧𝐴))
9 eqimss 3957 . . . . . . . . . . 11 (𝑧 = 𝐴𝑧𝐴)
109a1i 11 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧 = 𝐴𝑧𝐴))
118, 10jaod 859 . . . . . . . . 9 (𝐴 ∈ On → ((𝑧𝐴𝑧 = 𝐴) → 𝑧𝐴))
127, 11syl5 34 . . . . . . . 8 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴𝑧𝐴))
13 sseq2 3927 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑧𝑤𝑧𝐴))
1413rspcev 3537 . . . . . . . 8 ((𝐴 ∈ {𝐴} ∧ 𝑧𝐴) → ∃𝑤 ∈ {𝐴}𝑧𝑤)
156, 12, 14syl6an 684 . . . . . . 7 (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → ∃𝑤 ∈ {𝐴}𝑧𝑤))
1615ralrimiv 3104 . . . . . 6 (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)
17 ssun2 4087 . . . . . . 7 {𝐴} ⊆ (𝐴 ∪ {𝐴})
18 df-suc 6219 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
1917, 18sseqtrri 3938 . . . . . 6 {𝐴} ⊆ suc 𝐴
2016, 19jctil 523 . . . . 5 (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
21 snex 5324 . . . . . 6 {𝐴} ∈ V
22 fveq2 6717 . . . . . . . 8 (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
2322eqeq2d 2748 . . . . . . 7 (𝑦 = {𝐴} → (1o = (card‘𝑦) ↔ 1o = (card‘{𝐴})))
24 sseq1 3926 . . . . . . . 8 (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25 rexeq 3320 . . . . . . . . 9 (𝑦 = {𝐴} → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧𝑤))
2625ralbidv 3118 . . . . . . . 8 (𝑦 = {𝐴} → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))
2724, 26anbi12d 634 . . . . . . 7 (𝑦 = {𝐴} → ((𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤) ↔ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)))
2823, 27anbi12d 634 . . . . . 6 (𝑦 = {𝐴} → ((1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (1o = (card‘{𝐴}) ∧ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤))))
2921, 28spcev 3521 . . . . 5 ((1o = (card‘{𝐴}) ∧ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ {𝐴}𝑧𝑤)) → ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
305, 20, 29syl2anc 587 . . . 4 (𝐴 ∈ On → ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
31 1oex 8215 . . . . 5 1o ∈ V
32 eqeq1 2741 . . . . . . 7 (𝑥 = 1o → (𝑥 = (card‘𝑦) ↔ 1o = (card‘𝑦)))
3332anbi1d 633 . . . . . 6 (𝑥 = 1o → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
3433exbidv 1929 . . . . 5 (𝑥 = 1o → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
3531, 34elab 3587 . . . 4 (1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
3630, 35sylibr 237 . . 3 (𝐴 ∈ On → 1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
37 el1o 8226 . . . . 5 (𝑣 ∈ 1o𝑣 = ∅)
38 eqcom 2744 . . . . . . . . . . . . . . 15 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
39 vex 3412 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
40 onssnum 9654 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
4139, 40mpan 690 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ On → 𝑦 ∈ dom card)
42 cardnueq0 9580 . . . . . . . . . . . . . . . 16 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4341, 42syl 17 . . . . . . . . . . . . . . 15 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4438, 43syl5bb 286 . . . . . . . . . . . . . 14 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4544biimpa 480 . . . . . . . . . . . . 13 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
46 rex0 4272 . . . . . . . . . . . . . . . . 17 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4746a1i 11 . . . . . . . . . . . . . . . 16 (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
4847nrex 3188 . . . . . . . . . . . . . . 15 ¬ ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
49 nsuceq0 6293 . . . . . . . . . . . . . . . 16 suc 𝐴 ≠ ∅
50 r19.2z 4406 . . . . . . . . . . . . . . . 16 ((suc 𝐴 ≠ ∅ ∧ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5149, 50mpan 690 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤 → ∃𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤)
5248, 51mto 200 . . . . . . . . . . . . . 14 ¬ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤
53 rexeq 3320 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
5453ralbidv 3118 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧 ∈ suc 𝐴𝑤 ∈ ∅ 𝑧𝑤))
5552, 54mtbiri 330 . . . . . . . . . . . . 13 (𝑦 = ∅ → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5645, 55syl 17 . . . . . . . . . . . 12 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)
5756intnand 492 . . . . . . . . . . 11 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
58 imnan 403 . . . . . . . . . . 11 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
5957, 58mpbi 233 . . . . . . . . . 10 ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
60 suceloni 7592 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → suc 𝐴 ∈ On)
61 onss 7568 . . . . . . . . . . . . . . . . 17 (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
62 sstr 3909 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ⊆ On) → 𝑦 ⊆ On)
6361, 62sylan2 596 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ∈ On) → 𝑦 ⊆ On)
6460, 63sylan2 596 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ suc 𝐴𝐴 ∈ On) → 𝑦 ⊆ On)
6564ancoms 462 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴) → 𝑦 ⊆ On)
6665adantrr 717 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
67663adant2 1133 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
68 simp2 1139 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
69 simp3 1140 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))
7067, 68, 69jca31 518 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
71703expib 1124 . . . . . . . . . 10 (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7259, 71mtoi 202 . . . . . . . . 9 (𝐴 ∈ On → ¬ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
7372nexdv 1944 . . . . . . . 8 (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
74 0ex 5200 . . . . . . . . 9 ∅ ∈ V
75 eqeq1 2741 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
7675anbi1d 633 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7776exbidv 1929 . . . . . . . . 9 (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))))
7874, 77elab 3587 . . . . . . . 8 (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)))
7973, 78sylnibr 332 . . . . . . 7 (𝐴 ∈ On → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8079adantr 484 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
81 eleq1 2825 . . . . . . 7 (𝑣 = ∅ → (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}))
8281adantl 485 . . . . . 6 ((𝐴 ∈ On ∧ 𝑣 = ∅) → (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}))
8380, 82mtbird 328 . . . . 5 ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8437, 83sylan2b 597 . . . 4 ((𝐴 ∈ On ∧ 𝑣 ∈ 1o) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
8584ralrimiva 3105 . . 3 (𝐴 ∈ On → ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
86 cardon 9560 . . . . . . . 8 (card‘𝑦) ∈ On
87 eleq1 2825 . . . . . . . 8 (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
8886, 87mpbiri 261 . . . . . . 7 (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
8988adantr 484 . . . . . 6 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9089exlimiv 1938 . . . . 5 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤)) → 𝑥 ∈ On)
9190abssi 3983 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
92 oneqmini 6264 . . . 4 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ⊆ On → ((1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ∧ ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}) → 1o = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}))
9391, 92ax-mp 5 . . 3 ((1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))} ∧ ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))}) → 1o = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
9436, 85, 93syl2anc 587 . 2 (𝐴 ∈ On → 1o = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴𝑤𝑦 𝑧𝑤))})
953, 94eqtr4d 2780 1 (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847  w3a 1089   = wceq 1543  wex 1787  wcel 2110  {cab 2714  wne 2940  wral 3061  wrex 3062  Vcvv 3408  cun 3864  wss 3866  c0 4237  {csn 4541   cint 4859  dom cdm 5551  Oncon0 6213  suc csuc 6215  cfv 6380  1oc1o 8195  cardccrd 9551  cfccf 9553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-om 7645  df-wrecs 8047  df-recs 8108  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-cf 9557
This theorem is referenced by:  cflim2  9877  cfpwsdom  10198  rankcf  10391
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