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Theorem cfeq0 9685
 Description: Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
Assertion
Ref Expression
cfeq0 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))

Proof of Theorem cfeq0
Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 9676 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21eqeq1d 2800 . . 3 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅))
3 vex 3445 . . . . . . . . 9 𝑣 ∈ V
4 eqeq1 2802 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
54anbi1d 632 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
65exbidv 1922 . . . . . . . . 9 (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
73, 6elab 3616 . . . . . . . 8 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
8 fveq2 6655 . . . . . . . . . . . 12 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
9 cardidm 9390 . . . . . . . . . . . 12 (card‘(card‘𝑦)) = (card‘𝑦)
108, 9eqtrdi 2849 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
11 eqeq2 2810 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
1210, 11mpbird 260 . . . . . . . . . 10 (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
1312adantr 484 . . . . . . . . 9 ((𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
1413exlimiv 1931 . . . . . . . 8 (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
157, 14sylbi 220 . . . . . . 7 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘𝑣) = 𝑣)
16 cardon 9375 . . . . . . 7 (card‘𝑣) ∈ On
1715, 16eqeltrrdi 2899 . . . . . 6 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝑣 ∈ On)
1817ssriv 3921 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
19 onint0 7504 . . . . 5 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On → ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))}))
2018, 19ax-mp 5 . . . 4 ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 0ex 5179 . . . . . 6 ∅ ∈ V
22 eqeq1 2802 . . . . . . . 8 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
2322anbi1d 632 . . . . . . 7 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2423exbidv 1922 . . . . . 6 (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2521, 24elab 3616 . . . . 5 (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
26 onss 7498 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
27 sstr 3925 . . . . . . . . . . . 12 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2827ancoms 462 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2926, 28sylan 583 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
30293adant2 1128 . . . . . . . . 9 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ 𝑦𝐴) → 𝑦 ⊆ On)
31303adant3r 1178 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
32 simp2 1134 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
33 simp3 1135 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
34 eqcom 2805 . . . . . . . . . . . 12 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
35 vex 3445 . . . . . . . . . . . . . 14 𝑦 ∈ V
36 onssnum 9469 . . . . . . . . . . . . . 14 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
3735, 36mpan 689 . . . . . . . . . . . . 13 (𝑦 ⊆ On → 𝑦 ∈ dom card)
38 cardnueq0 9395 . . . . . . . . . . . . 13 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
3937, 38syl 17 . . . . . . . . . . . 12 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4034, 39syl5bb 286 . . . . . . . . . . 11 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4140biimpa 480 . . . . . . . . . 10 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
42 sseq1 3942 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ⊆ 𝐴))
43 rexeq 3360 . . . . . . . . . . . . 13 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
4443ralbidv 3162 . . . . . . . . . . . 12 (𝑦 = ∅ → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4542, 44anbi12d 633 . . . . . . . . . . 11 (𝑦 = ∅ → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)))
4645biimpa 480 . . . . . . . . . 10 ((𝑦 = ∅ ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4741, 46sylan 583 . . . . . . . . 9 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
48 rex0 4274 . . . . . . . . . . . . 13 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4948rgenw 3118 . . . . . . . . . . . 12 𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
50 r19.2z 4401 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
5149, 50mpan2 690 . . . . . . . . . . 11 (𝐴 ≠ ∅ → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
52 rexnal 3201 . . . . . . . . . . 11 (∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤 ↔ ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5351, 52sylib 221 . . . . . . . . . 10 (𝐴 ≠ ∅ → ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5453necon4ai 3018 . . . . . . . . 9 (∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤𝐴 = ∅)
5547, 54simpl2im 507 . . . . . . . 8 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
5631, 32, 33, 55syl21anc 836 . . . . . . 7 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
57563expib 1119 . . . . . 6 (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5857exlimdv 1934 . . . . 5 (𝐴 ∈ On → (∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5925, 58syl5bi 245 . . . 4 (𝐴 ∈ On → (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝐴 = ∅))
6020, 59syl5bi 245 . . 3 (𝐴 ∈ On → ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ → 𝐴 = ∅))
612, 60sylbid 243 . 2 (𝐴 ∈ On → ((cf‘𝐴) = ∅ → 𝐴 = ∅))
62 fveq2 6655 . . 3 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
63 cf0 9680 . . 3 (cf‘∅) = ∅
6462, 63eqtrdi 2849 . 2 (𝐴 = ∅ → (cf‘𝐴) = ∅)
6561, 64impbid1 228 1 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3442   ⊆ wss 3883  ∅c0 4246  ∩ cint 4842  dom cdm 5523  Oncon0 6166  ‘cfv 6332  cardccrd 9366  cfccf 9368 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-wrecs 7948  df-recs 8009  df-er 8290  df-en 8511  df-dom 8512  df-card 9370  df-cf 9372 This theorem is referenced by: (None)
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