Theorem grothomex 10917

Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9647). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) Use omex 9647 instead. (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 9785	. . . 4 ⊢ 𝑅1:On–1-1→V
2		omsson 7881	. . . 4 ⊢ ω ⊆ On
3		f1ores 6839	. . . 4 ⊢ ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω))
4	1, 2, 3	mp2an 705	. . 3 ⊢ (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω)
5		f1of1 6823	. . 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 9776	. . . . . . . 8 ⊢ 𝑅1 Fn On
8		fvelimab 6957	. . . . . . . 8 ⊢ ((𝑅1 Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤))
9	7, 2, 8	mp2an 705	. . . . . . 7 ⊢ (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤)
10		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
11	10	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑅1‘𝑥) = (𝑅1‘𝑤))
13	12	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘𝑤) ∈ 𝑦))
14		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑤 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑤))
15	14	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16		r10 9778	. . . . . . . . . . . 12 ⊢ (𝑅1‘∅) = ∅
17	16	eleq1i 2851	. . . . . . . . . . 11 ⊢ ((𝑅1‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
18	17	biranri 511	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1‘∅) ∈ 𝑦)
19		pweq 4570	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1‘𝑤))
20	19	eleq1d 2845	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑤) → (𝒫 𝑧 ∈ 𝑦 ↔ 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
21	20	rspccv 3573	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
22		nnon 7883	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
23		r1suc 9780	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ On → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1‘𝑤))
24	22, 23	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ω → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1‘𝑤))
25	24	eleq1d 2845	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ω → ((𝑅1‘suc 𝑤) ∈ 𝑦 ↔ 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
26	25	biimprcd 253	. . . . . . . . . . . . 13 ⊢ (𝒫 (𝑅1‘𝑤) ∈ 𝑦 → (𝑤 ∈ ω → (𝑅1‘suc 𝑤) ∈ 𝑦))
27	21, 26	syl6 36	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑤 ∈ ω → (𝑅1‘suc 𝑤) ∈ 𝑦)))
28	27	com3r 88	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
29	28	adantld 496	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
30	11, 13, 15, 18, 29	finds2 7910	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1‘𝑥) ∈ 𝑦))
31		eleq1 2848	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
32	31	biimpd 232	. . . . . . . . 9 ⊢ ((𝑅1‘𝑥) = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 → 𝑤 ∈ 𝑦))
33	30, 32	syl9 78	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦)))
34	33	rexlimiv 3156	. . . . . . 7 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦))
35	9, 34	sylbi 220	. . . . . 6 ⊢ (𝑤 ∈ (𝑅1 “ ω) → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦))
36	35	com12 33	. . . . 5 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑤 ∈ (𝑅1 “ ω) → 𝑤 ∈ 𝑦))
37	36	ssrdv 3914	. . . 4 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
38		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
39	38	ssex 5281	. . . 4 ⊢ ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
40	37, 39	syl 18	. . 3 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ∈ V)
41		0ex 5260	. . . 4 ⊢ ∅ ∈ V
42		eleq1 2848	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
43	42	anbi1d 643	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
44	43	exbidv 1953	. . . 4 ⊢ (𝑥 = ∅ → (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
45		axgroth6 10916	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))
46		simpr 490	. . . . . . . 8 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → 𝒫 𝑧 ∈ 𝑦)
47	46	ralimi 3099	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
48	47	anim2i 629	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦)) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
49	48	3adant3 1150	. . . . 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))
50	45, 49	eximii 1869	. . . 4 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
51	41, 44, 50	vtocl 3520	. . 3 ⊢ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
52	40, 51	exlimiiv 1963	. 2 ⊢ (𝑅1 “ ω) ∈ V
53		f1dmex 7969	. 2 ⊢ (((𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω) ∧ (𝑅1 “ ω) ∈ V) → ω ∈ V)
54	6, 52, 53	mp2an 705	1 ⊢ ω ∈ V

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556   class class class wbr 5102   ↾ cres 5652   “ cima 5653  Oncon0 6362  suc csuc 6364   Fn wfn 6533  –1-1→wf1 6535  –1-1-onto→wf1o 6537  ‘cfv 6538  ωcom 7877   ≺ csdm 8975  𝑅₁cr1 9769

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-groth 10911

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-r1 9771

This theorem is used by: (None)