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Theorem dominf 9150
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 9140. See dominfac 9274 for a version proved from ax-ac 9164. The axiom of Regularity is used for this proof, via inf3lem6 8413, 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 2844 . . . 4 (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅))
3 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
4 unieq 4380 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
53, 4sseq12d 3597 . . . 4 (𝑥 = 𝐴 → (𝑥 𝑥𝐴 𝐴))
62, 5anbi12d 743 . . 3 (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 𝐴)))
7 breq2 4587 . . 3 (𝑥 = 𝐴 → (ω ≼ 𝑥 ↔ ω ≼ 𝐴))
86, 7imbi12d 333 . 2 (𝑥 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ≼ 𝑥) ↔ ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)))
9 eqid 2610 . . . 4 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
10 eqid 2610 . . . 4 (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω)
119, 10, 1, 1inf3lem6 8413 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥)
12 vpwex 4775 . . . 4 𝒫 𝑥 ∈ V
1312f1dom 7863 . . 3 ((rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥)
14 pwfi 8144 . . . . . . 7 (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin)
1514biimpi 205 . . . . . 6 (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin)
16 isfinite 8432 . . . . . 6 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
17 isfinite 8432 . . . . . 6 (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω)
1815, 16, 173imtr3i 279 . . . . 5 (𝑥 ≺ ω → 𝒫 𝑥 ≺ ω)
1918con3i 149 . . . 4 (¬ 𝒫 𝑥 ≺ ω → ¬ 𝑥 ≺ ω)
2012domtriom 9148 . . . 4 (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω)
21 vex 3176 . . . . 5 𝑥 ∈ V
2221domtriom 9148 . . . 4 (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
2319, 20, 223imtr4i 280 . . 3 (ω ≼ 𝒫 𝑥 → ω ≼ 𝑥)
2411, 13, 233syl 18 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ≼ 𝑥)
251, 8, 24vtocl 3232 1 ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  {crab 2900  Vcvv 3173  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108   cuni 4372   class class class wbr 4583  cmpt 4643  cres 5040  1-1wf1 5801  ωcom 6957  reccrdg 7392  cdom 7839  csdm 7840  Fincfn 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421  ax-cc 9140
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873
This theorem is referenced by:  axgroth3  9532
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