Theorem grothomex 10740

Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9552). 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 9687	. . . 4 ⊢ 𝑅1:On–1-1→V
2		omsson 7812	. . . 4 ⊢ ω ⊆ On
3		f1ores 6788	. . . 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 6773	. . 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 9679	. . . . . . . 8 ⊢ 𝑅1 Fn On
8		fvelimab 6906	. . . . . . . 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 6834	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
11	10	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑅1‘𝑥) = (𝑅1‘𝑤))
13	12	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘𝑤) ∈ 𝑦))
14		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑤 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑤))
15	14	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16		r10 9680	. . . . . . . . . . . . 13 ⊢ (𝑅1‘∅) = ∅
17	16	eleq1i 2827	. . . . . . . . . . . 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 4568	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1‘𝑤))
21	20	eleq1d 2821	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑤) → (𝒫 𝑧 ∈ 𝑦 ↔ 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
22	21	rspccv 3573	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
23		nnon 7814	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
24		r1suc 9682	. . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . 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 7840	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1‘𝑥) ∈ 𝑦))
32		eleq1 2824	. . . . . . . . . 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 3130	. . . . . . 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 3939	. . . 4 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
39		vex 3444	. . . . 5 ⊢ 𝑦 ∈ V
40	39	ssex 5266	. . . 4 ⊢ ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
41	38, 40	syl 17	. . 3 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ∈ V)
42		0ex 5252	. . . 4 ⊢ ∅ ∈ V
43		eleq1 2824	. . . . . 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 1922	. . . 4 ⊢ (𝑥 = ∅ → (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
46		axgroth6 10739	. . . . 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 3073	. . . . . . 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 1838	. . . 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 1932	. 2 ⊢ (𝑅1 “ ω) ∈ V
54		f1dmex 7901	. 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 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554   class class class wbr 5098   ↾ cres 5626   “ cima 5627  Oncon0 6317  suc csuc 6319   Fn wfn 6487  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492  ωcom 7808   ≺ csdm 8882  𝑅₁cr1 9674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-groth 10734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-r1 9676

This theorem is referenced by: (None)