Theorem dominf 9267

Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 9257. See dominfac 9395 for a version proved from ax-ac 9281. The axiom of Regularity is used for this proof, via inf3lem6 8530, 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 2856	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅))
3		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4		unieq 4444	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
5	3, 4	sseq12d 3634	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝐴))
6	2, 5	anbi12d 747	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴)))
7		breq2 4657	. . 3 ⊢ (𝑥 = 𝐴 → (ω ≼ 𝑥 ↔ ω ≼ 𝐴))
8	6, 7	imbi12d 334	. 2 ⊢ (𝑥 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) ↔ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴)))
9		eqid 2622	. . . 4 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
10		eqid 2622	. . . 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 8530	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥)
12		vpwex 4849	. . . 4 ⊢ 𝒫 𝑥 ∈ V
13	12	f1dom 7977	. . 3 ⊢ ((rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥)
14		pwfi 8261	. . . . . . 7 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin)
15	14	biimpi 206	. . . . . 6 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin)
16		isfinite 8549	. . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
17		isfinite 8549	. . . . . 6 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω)
18	15, 16, 17	3imtr3i 280	. . . . 5 ⊢ (𝑥 ≺ ω → 𝒫 𝑥 ≺ ω)
19	18	con3i 150	. . . 4 ⊢ (¬ 𝒫 𝑥 ≺ ω → ¬ 𝑥 ≺ ω)
20	12	domtriom 9265	. . . 4 ⊢ (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω)
21		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
22	21	domtriom 9265	. . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
23	19, 20, 22	3imtr4i 281	. . 3 ⊢ (ω ≼ 𝒫 𝑥 → ω ≼ 𝑥)
24	11, 13, 23	3syl 18	. 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥)
25	1, 8, 24	vtocl 3259	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   ↾ cres 5116  –1-1→wf1 5885  ωcom 7065  reccrdg 7505   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538  ax-cc 9257

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990

This theorem is referenced by:  axgroth3  9653