Theorem domtriom 9859

Description: Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 9730) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)

Hypothesis

Ref	Expression
domtriom.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
domtriom	⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)

Proof of Theorem domtriom

Dummy variables 𝑏 𝑛 𝑦 𝑗 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		domnsym 8637	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ≺ ω)
2		isfinite 9109	. . 3 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
3		domtriom.1	. . . 4 ⊢ 𝐴 ∈ V
4		eqid 2821	. . . 4 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} = {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)}
5		fveq2 6665	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑏‘𝑚) = (𝑏‘𝑛))
6		fveq2 6665	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑏‘𝑗) = (𝑏‘𝑘))
7	6	cbviunv 4958	. . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘)
8		iuneq1 4928	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))
9	7, 8	syl5eq 2868	. . . . . 6 ⊢ (𝑚 = 𝑛 → ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))
10	5, 9	difeq12d 4100	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗)) = ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)))
11	10	cbvmptv 5162	. . . 4 ⊢ (𝑚 ∈ ω ↦ ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗))) = (𝑛 ∈ ω ↦ ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)))
12	3, 4, 11	domtriomlem 9858	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ω ≼ 𝐴)
13	2, 12	sylnbir 333	. 2 ⊢ (¬ 𝐴 ≺ ω → ω ≼ 𝐴)
14	1, 13	impbii 211	1 ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  {cab 2799  Vcvv 3495   ∖ cdif 3933   ⊆ wss 3936  𝒫 cpw 4539  ∪ ciun 4912   class class class wbr 5059   ↦ cmpt 5139  ‘cfv 6350  ωcom 7574   ≈ cen 8500   ≼ cdom 8501   ≺ csdm 8502  Fincfn 8503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-cc 9851

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362

This theorem is referenced by:  fin41  9860  dominf  9861