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Theorem dominf 10477
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 10467. See dominfac 10605 for a version proved from ax-ac 10491. The axiom of Regularity is used for this proof, via inf3lem6 9667, and its use is necessary: otherwise the set 𝐴 = {𝐴} or 𝐴 = {∅, 𝐴} (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
Hypothesis
Ref Expression
dominf.1 𝐴 ∈ V
Assertion
Ref Expression
dominf ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)

Proof of Theorem dominf
Dummy variables 𝑥 𝑦 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dominf.1 . 2 𝐴 ∈ V
2 neeq1 2993 . . . 4 (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅))
3 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
4 unieq 4917 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
53, 4sseq12d 4013 . . . 4 (𝑥 = 𝐴 → (𝑥 𝑥𝐴 𝐴))
62, 5anbi12d 630 . . 3 (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 𝐴)))
7 breq2 5148 . . 3 (𝑥 = 𝐴 → (ω ≼ 𝑥 ↔ ω ≼ 𝐴))
86, 7imbi12d 343 . 2 (𝑥 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ≼ 𝑥) ↔ ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)))
9 eqid 2726 . . . 4 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
10 eqid 2726 . . . 4 (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω)
119, 10, 1, 1inf3lem6 9667 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥)
12 vpwex 5372 . . . 4 𝒫 𝑥 ∈ V
1312f1dom 8995 . . 3 ((rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥)
14 pwfi 9350 . . . . . . 7 (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin)
1514biimpi 215 . . . . . 6 (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin)
16 isfinite 9686 . . . . . 6 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
17 isfinite 9686 . . . . . 6 (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω)
1815, 16, 173imtr3i 290 . . . . 5 (𝑥 ≺ ω → 𝒫 𝑥 ≺ ω)
1918con3i 154 . . . 4 (¬ 𝒫 𝑥 ≺ ω → ¬ 𝑥 ≺ ω)
2012domtriom 10475 . . . 4 (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω)
21 vex 3467 . . . . 5 𝑥 ∈ V
2221domtriom 10475 . . . 4 (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
2319, 20, 223imtr4i 291 . . 3 (ω ≼ 𝒫 𝑥 → ω ≼ 𝑥)
2411, 13, 233syl 18 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ≼ 𝑥)
251, 8, 24vtocl 3538 1 ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wcel 2099  wne 2930  {crab 3420  Vcvv 3463  cin 3946  wss 3947  c0 4323  𝒫 cpw 4598   cuni 4906   class class class wbr 5144  cmpt 5227  cres 5675  1-1wf1 6541  ωcom 7866  reccrdg 8429  cdom 8962  csdm 8963  Fincfn 8964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-reg 9626  ax-inf2 9675  ax-cc 10467
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-oadd 8490  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9935  df-card 9973
This theorem is referenced by:  axgroth3  10863
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