Theorem grothomex 10784

Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9595). 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 9730	. . . 4 ⊢ 𝑅1:On–1-1→V
2		omsson 7846	. . . 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 702	. . 3 ⊢ (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω)
5		f1of1 6801	. . 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 9722	. . . . . . . 8 ⊢ 𝑅1 Fn On
8		fvelimab 6935	. . . . . . . 8 ⊢ ((𝑅1 Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤))
9	7, 2, 8	mp2an 702	. . . . . . 7 ⊢ (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤)
10		fveq2 6863	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
11	10	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12		fveq2 6863	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑅1‘𝑥) = (𝑅1‘𝑤))
13	12	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘𝑤) ∈ 𝑦))
14		fveq2 6863	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑤 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑤))
15	14	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16		r10 9723	. . . . . . . . . . . 12 ⊢ (𝑅1‘∅) = ∅
17	16	eleq1i 2852	. . . . . . . . . . 11 ⊢ ((𝑅1‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
18	17	biranri 509	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1‘∅) ∈ 𝑦)
19		pweq 4568	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1‘𝑤))
20	19	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑤) → (𝒫 𝑧 ∈ 𝑦 ↔ 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
21	20	rspccv 3578	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
22		nnon 7848	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
23		r1suc 9725	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ On → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1‘𝑤))
24	22, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ω → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1‘𝑤))
25	24	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ω → ((𝑅1‘suc 𝑤) ∈ 𝑦 ↔ 𝒫 (𝑅1‘𝑤) ∈ 𝑦))
26	25	biimprcd 252	. . . . . . . . . . . . 13 ⊢ (𝒫 (𝑅1‘𝑤) ∈ 𝑦 → (𝑤 ∈ ω → (𝑅1‘suc 𝑤) ∈ 𝑦))
27	21, 26	syl6 35	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑤 ∈ ω → (𝑅1‘suc 𝑤) ∈ 𝑦)))
28	27	com3r 87	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
29	28	adantld 494	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → ((𝑅1‘𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
30	11, 13, 15, 18, 29	finds2 7875	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1‘𝑥) ∈ 𝑦))
31		eleq1 2849	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑥) = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
32	31	biimpd 231	. . . . . . . . 9 ⊢ ((𝑅1‘𝑥) = 𝑤 → ((𝑅1‘𝑥) ∈ 𝑦 → 𝑤 ∈ 𝑦))
33	30, 32	syl9 77	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦)))
34	33	rexlimiv 3155	. . . . . . 7 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦))
35	9, 34	sylbi 219	. . . . . 6 ⊢ (𝑤 ∈ (𝑅1 “ ω) → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦))
36	35	com12 32	. . . . 5 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑤 ∈ (𝑅1 “ ω) → 𝑤 ∈ 𝑦))
37	36	ssrdv 3942	. . . 4 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
38		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
39	38	ssex 5276	. . . 4 ⊢ ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
40	37, 39	syl 17	. . 3 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅1 “ ω) ∈ V)
41		0ex 5256	. . . 4 ⊢ ∅ ∈ V
42		eleq1 2849	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
43	42	anbi1d 640	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
44	43	exbidv 1940	. . . 4 ⊢ (𝑥 = ∅ → (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
45		axgroth6 10783	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))
46		simpr 488	. . . . . . . 8 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → 𝒫 𝑧 ∈ 𝑦)
47	46	ralimi 3098	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
48	47	anim2i 626	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦)) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
49	48	3adant3 1144	. . . . 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 1856	. . . 4 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
51	41, 44, 50	vtocl 3524	. . 3 ⊢ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
52	40, 51	exlimiiv 1950	. 2 ⊢ (𝑅1 “ ω) ∈ V
53		f1dmex 7934	. 2 ⊢ (((𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω) ∧ (𝑅1 “ ω) ∈ V) → ω ∈ V)
54	6, 52, 53	mp2an 702	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554   class class class wbr 5099   ↾ cres 5647   “ cima 5648  Oncon0 6342  suc csuc 6344   Fn wfn 6512  –1-1→wf1 6514  –1-1-onto→wf1o 6516  ‘cfv 6517  ωcom 7842   ≺ csdm 8922  𝑅₁cr1 9717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-groth 10778

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-r1 9719

This theorem is referenced by: (None)