Theorem dominf 10398

Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 10388. See dominfac 10526 for a version proved from ax-ac 10412. The axiom of Regularity is used for this proof, via inf3lem6 9586, 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.

Step	Hyp	Ref	Expression
1		dominf.1	. 2 ⊢ 𝐴 ∈ V
2		neeq1 2987	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅))
3		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4		unieq 4882	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
5	3, 4	sseq12d 3980	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝐴))
6	2, 5	anbi12d 632	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴)))
7		breq2 5111	. . 3 ⊢ (𝑥 = 𝐴 → (ω ≼ 𝑥 ↔ ω ≼ 𝐴))
8	6, 7	imbi12d 344	. 2 ⊢ (𝑥 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) ↔ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴)))
9		eqid 2729	. . . 4 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
10		eqid 2729	. . . 4 ⊢ (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω)
11	9, 10, 1, 1	inf3lem6 9586	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥)
12		vpwex 5332	. . . 4 ⊢ 𝒫 𝑥 ∈ V
13	12	f1dom 8945	. . 3 ⊢ ((rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥)
14		pwfi 9268	. . . . . . 7 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin)
15	14	biimpi 216	. . . . . 6 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin)
16		isfinite 9605	. . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
17		isfinite 9605	. . . . . 6 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω)
18	15, 16, 17	3imtr3i 291	. . . . 5 ⊢ (𝑥 ≺ ω → 𝒫 𝑥 ≺ ω)
19	18	con3i 154	. . . 4 ⊢ (¬ 𝒫 𝑥 ≺ ω → ¬ 𝑥 ≺ ω)
20	12	domtriom 10396	. . . 4 ⊢ (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω)
21		vex 3451	. . . . 5 ⊢ 𝑥 ∈ V
22	21	domtriom 10396	. . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
23	19, 20, 22	3imtr4i 292	. . 3 ⊢ (ω ≼ 𝒫 𝑥 → ω ≼ 𝑥)
24	11, 13, 23	3syl 18	. 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥)
25	1, 8, 24	vtocl 3524	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3405  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  ∪ cuni 4871   class class class wbr 5107   ↦ cmpt 5188   ↾ cres 5640  –1-1→wf1 6508  ωcom 7842  reccrdg 8377   ≼ cdom 8916   ≺ csdm 8917  Fincfn 8918

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-reg 9545  ax-inf2 9594  ax-cc 10388

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892

This theorem is referenced by:  axgroth3  10784