Theorem domtriom 9865

Description: Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 9736) 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 8643	. 2 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ≺ ω)
2		isfinite 9115	. . 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 6670	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑏‘𝑚) = (𝑏‘𝑛))
6		fveq2 6670	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑏‘𝑗) = (𝑏‘𝑘))
7	6	cbviunv 4965	. . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘)
8		iuneq1 4935	. . . . . . 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 5169	. . . 4 ⊢ (𝑚 ∈ ω ↦ ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗))) = (𝑛 ∈ ω ↦ ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)))
12	3, 4, 11	domtriomlem 9864	. . 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 2114  {cab 2799  Vcvv 3494   ∖ cdif 3933   ⊆ wss 3936  𝒫 cpw 4539  ∪ ciun 4919   class class class wbr 5066   ↦ cmpt 5146  ‘cfv 6355  ωcom 7580   ≈ cen 8506   ≼ cdom 8507   ≺ csdm 8508  Fincfn 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cc 9857

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3496  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 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-dju 9330  df-card 9368

This theorem is referenced by:  fin41  9866  dominf  9867