Theorem grothomex 10742

Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9558). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
grothomex	⊢ ω ∈ V

Proof of Theorem grothomex

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r111 9690	. . . 4 ⊢ 𝑅1:On–1-1→V
2		omsson 7810	. . . 4 ⊢ ω ⊆ On
3		f1ores 6782	. . . 4 ⊢ ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω))
4	1, 2, 3	mp2an 692	. . 3 ⊢ (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω)
5		f1of1 6767	. . 3 ⊢ ((𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω) → (𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω))
6	4, 5	ax-mp 5	. 2 ⊢ (𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω)
7		r1fnon 9682	. . . . . . . 8 ⊢ 𝑅1 Fn On
8		fvelimab 6899	. . . . . . . 8 ⊢ ((𝑅1 Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤))
9	7, 2, 8	mp2an 692	. . . . . . 7 ⊢ (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤)
10		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
11	10	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑅1‘𝑥) = (𝑅1‘𝑤))
13	12	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘𝑤) ∈ 𝑦))
14		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑤 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑤))
15	14	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16		r10 9683	. . . . . . . . . . . . 13 ⊢ (𝑅1‘∅) = ∅
17	16	eleq1i 2819	. . . . . . . . . . . 12 ⊢ ((𝑅1‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
18	17	biimpri 228	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝑦 → (𝑅1‘∅) ∈ 𝑦)
19	18	adantr 480	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1‘∅) ∈ 𝑦)
20		pweq 4567	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1‘𝑤))
21	20	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑤) → (𝒫 𝑧 ∈ 𝑦 ↔ 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
22	21	rspccv 3576	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
23		nnon 7812	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
24		r1suc 9685	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ On → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1‘𝑤))
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ω → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1‘𝑤))
26	25	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ω → ((𝑅1‘suc 𝑤) ∈ 𝑦 ↔ 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
27	26	biimprcd 250	. . . . . . . . . . . . 13 ⊢ (𝒫 (𝑅1‘𝑤) ∈ 𝑦 → (𝑤 ∈ ω → (𝑅1‘suc 𝑤) ∈ 𝑦))
28	22, 27	syl6 35	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑤 ∈ ω → (𝑅1‘suc 𝑤) ∈ 𝑦)))
29	28	com3r 87	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
30	29	adantld 490	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
31	11, 13, 15, 19, 30	finds2 7838	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1‘𝑥) ∈ 𝑦))
32		eleq1 2816	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
33	32	biimpd 229	. . . . . . . . 9 ⊢ ((𝑅1‘𝑥) = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 → 𝑤 ∈ 𝑦))
34	31, 33	syl9 77	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦)))
35	34	rexlimiv 3123	. . . . . . 7 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦))
36	9, 35	sylbi 217	. . . . . 6 ⊢ (𝑤 ∈ (𝑅1 “ ω) → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦))
37	36	com12 32	. . . . 5 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑤 ∈ (𝑅1 “ ω) → 𝑤 ∈ 𝑦))
38	37	ssrdv 3943	. . . 4 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
39		vex 3442	. . . . 5 ⊢ 𝑦 ∈ V
40	39	ssex 5263	. . . 4 ⊢ ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
41	38, 40	syl 17	. . 3 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ∈ V)
42		0ex 5249	. . . 4 ⊢ ∅ ∈ V
43		eleq1 2816	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
44	43	anbi1d 631	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
45	44	exbidv 1921	. . . 4 ⊢ (𝑥 = ∅ → (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
46		axgroth6 10741	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))
47		simpr 484	. . . . . . . 8 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → 𝒫 𝑧 ∈ 𝑦)
48	47	ralimi 3066	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
49	48	anim2i 617	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦)) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
50	49	3adant3 1132	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
51	46, 50	eximii 1837	. . . 4 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
52	42, 45, 51	vtocl 3515	. . 3 ⊢ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
53	41, 52	exlimiiv 1931	. 2 ⊢ (𝑅1 “ ω) ∈ V
54		f1dmex 7899	. 2 ⊢ (((𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω) ∧ (𝑅1 “ ω) ∈ V) → ω ∈ V)
55	6, 53, 54	mp2an 692	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3438   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4553   class class class wbr 5095   ↾ cres 5625   “ cima 5626  Oncon0 6311  suc csuc 6313   Fn wfn 6481  –1-1→wf1 6483  –1-1-onto→wf1o 6485  ‘cfv 6486  ωcom 7806   ≺ csdm 8878  𝑅₁cr1 9677

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-groth 10736

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-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-r1 9679

This theorem is referenced by: (None)