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Theorem cfeq0 10296
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 10287 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21eqeq1d 2739 . . 3 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅))
3 vex 3484 . . . . . . . . 9 𝑣 ∈ V
4 eqeq1 2741 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
54anbi1d 631 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
65exbidv 1921 . . . . . . . . 9 (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
73, 6elab 3679 . . . . . . . 8 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
8 fveq2 6906 . . . . . . . . . . . 12 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
9 cardidm 9999 . . . . . . . . . . . 12 (card‘(card‘𝑦)) = (card‘𝑦)
108, 9eqtrdi 2793 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
11 eqeq2 2749 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
1210, 11mpbird 257 . . . . . . . . . 10 (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
1312adantr 480 . . . . . . . . 9 ((𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
1413exlimiv 1930 . . . . . . . 8 (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
157, 14sylbi 217 . . . . . . 7 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘𝑣) = 𝑣)
16 cardon 9984 . . . . . . 7 (card‘𝑣) ∈ On
1715, 16eqeltrrdi 2850 . . . . . 6 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝑣 ∈ On)
1817ssriv 3987 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
19 onint0 7811 . . . . 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 5307 . . . . . 6 ∅ ∈ V
22 eqeq1 2741 . . . . . . . 8 (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
2322anbi1d 631 . . . . . . 7 (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2423exbidv 1921 . . . . . 6 (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2521, 24elab 3679 . . . . 5 (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
26 onss 7805 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
27 sstr 3992 . . . . . . . . . . . 12 ((𝑦𝐴𝐴 ⊆ On) → 𝑦 ⊆ On)
2827ancoms 458 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
2926, 28sylan 580 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ⊆ On)
30293adant2 1132 . . . . . . . . 9 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ 𝑦𝐴) → 𝑦 ⊆ On)
31303adant3r 1182 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝑦 ⊆ On)
32 simp2 1138 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∅ = (card‘𝑦))
33 simp3 1139 . . . . . . . 8 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
34 eqcom 2744 . . . . . . . . . . . 12 (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
35 vex 3484 . . . . . . . . . . . . . 14 𝑦 ∈ V
36 onssnum 10080 . . . . . . . . . . . . . 14 ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
3735, 36mpan 690 . . . . . . . . . . . . 13 (𝑦 ⊆ On → 𝑦 ∈ dom card)
38 cardnueq0 10004 . . . . . . . . . . . . 13 (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
3937, 38syl 17 . . . . . . . . . . . 12 (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
4034, 39bitrid 283 . . . . . . . . . . 11 (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
4140biimpa 476 . . . . . . . . . 10 ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
42 sseq1 4009 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝐴 ↔ ∅ ⊆ 𝐴))
43 rexeq 3322 . . . . . . . . . . . . 13 (𝑦 = ∅ → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧𝑤))
4443ralbidv 3178 . . . . . . . . . . . 12 (𝑦 = ∅ → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4542, 44anbi12d 632 . . . . . . . . . . 11 (𝑦 = ∅ → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)))
4645biimpa 476 . . . . . . . . . 10 ((𝑦 = ∅ ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
4741, 46sylan 580 . . . . . . . . 9 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤))
48 rex0 4360 . . . . . . . . . . . . 13 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
4948rgenw 3065 . . . . . . . . . . . 12 𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤
50 r19.2z 4495 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤) → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
5149, 50mpan2 691 . . . . . . . . . . 11 (𝐴 ≠ ∅ → ∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤)
52 rexnal 3100 . . . . . . . . . . 11 (∃𝑧𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧𝑤 ↔ ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5351, 52sylib 218 . . . . . . . . . 10 (𝐴 ≠ ∅ → ¬ ∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤)
5453necon4ai 2972 . . . . . . . . 9 (∀𝑧𝐴𝑤 ∈ ∅ 𝑧𝑤𝐴 = ∅)
5547, 54simpl2im 503 . . . . . . . 8 (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
5631, 32, 33, 55syl21anc 838 . . . . . . 7 ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅)
57563expib 1123 . . . . . 6 (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5857exlimdv 1933 . . . . 5 (𝐴 ∈ On → (∃𝑦(∅ = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → 𝐴 = ∅))
5925, 58biimtrid 242 . . . 4 (𝐴 ∈ On → (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝐴 = ∅))
6020, 59biimtrid 242 . . 3 (𝐴 ∈ On → ( {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} = ∅ → 𝐴 = ∅))
612, 60sylbid 240 . 2 (𝐴 ∈ On → ((cf‘𝐴) = ∅ → 𝐴 = ∅))
62 fveq2 6906 . . 3 (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
63 cf0 10291 . . 3 (cf‘∅) = ∅
6462, 63eqtrdi 2793 . 2 (𝐴 = ∅ → (cf‘𝐴) = ∅)
6561, 64impbid1 225 1 (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  Vcvv 3480  wss 3951  c0 4333   cint 4946  dom cdm 5685  Oncon0 6384  cfv 6561  cardccrd 9975  cfccf 9977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-er 8745  df-en 8986  df-dom 8987  df-card 9979  df-cf 9981
This theorem is referenced by: (None)
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