Theorem grothomex 10789

Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9603). 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 9735	. . . 4 ⊢ 𝑅1:On–1-1→V
2		omsson 7849	. . . 4 ⊢ ω ⊆ On
3		f1ores 6817	. . . 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 6802	. . 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 9727	. . . . . . . 8 ⊢ 𝑅1 Fn On
8		fvelimab 6936	. . . . . . . 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 6861	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
11	10	eleq1d 2814	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑅1‘𝑥) = (𝑅1‘𝑤))
13	12	eleq1d 2814	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘𝑤) ∈ 𝑦))
14		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑤 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑤))
15	14	eleq1d 2814	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16		r10 9728	. . . . . . . . . . . . 13 ⊢ (𝑅1‘∅) = ∅
17	16	eleq1i 2820	. . . . . . . . . . . 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 4580	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1‘𝑤))
21	20	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑤) → (𝒫 𝑧 ∈ 𝑦 ↔ 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
22	21	rspccv 3588	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
23		nnon 7851	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
24		r1suc 9730	. . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . 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 7877	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1‘𝑥) ∈ 𝑦))
32		eleq1 2817	. . . . . . . . . 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 3128	. . . . . . 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 3955	. . . 4 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
39		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
40	39	ssex 5279	. . . 4 ⊢ ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
41	38, 40	syl 17	. . 3 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ∈ V)
42		0ex 5265	. . . 4 ⊢ ∅ ∈ V
43		eleq1 2817	. . . . . 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 10788	. . . . 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 3067	. . . . . . 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 3527	. . 3 ⊢ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
53	41, 52	exlimiiv 1931	. 2 ⊢ (𝑅1 “ ω) ∈ V
54		f1dmex 7938	. 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 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566   class class class wbr 5110   ↾ cres 5643   “ cima 5644  Oncon0 6335  suc csuc 6337   Fn wfn 6509  –1-1→wf1 6511  –1-1-onto→wf1o 6513  ‘cfv 6514  ωcom 7845   ≺ csdm 8920  𝑅₁cr1 9722

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-groth 10783

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-r1 9724

This theorem is referenced by: (None)