grothomex - Metamath Proof Explorer

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Theorem grothomex 10824

Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9638). 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 9770	. . . 4 ⊢ 𝑅₁:On–1-1→V
2		omsson 7859	. . . 4 ⊢ ω ⊆ On
3		f1ores 6848	. . . 4 ⊢ ((𝑅₁:On–1-1→V ∧ ω ⊆ On) → (𝑅₁ ↾ ω):ω–1-1-onto→(𝑅₁ “ ω))
4	1, 2, 3	mp2an 691	. . 3 ⊢ (𝑅₁ ↾ ω):ω–1-1-onto→(𝑅₁ “ ω)
5		f1of1 6833	. . 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 9762	. . . . . . . 8 ⊢ 𝑅₁ Fn On
8		fvelimab 6965	. . . . . . . 8 ⊢ ((𝑅₁ Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅₁ “ ω) ↔ ∃𝑥 ∈ ω (𝑅₁‘𝑥) = 𝑤))
9	7, 2, 8	mp2an 691	. . . . . . 7 ⊢ (𝑤 ∈ (𝑅₁ “ ω) ↔ ∃𝑥 ∈ ω (𝑅₁‘𝑥) = 𝑤)
10		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑅₁‘𝑥) = (𝑅₁‘∅))
11	10	eleq1d 2819	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ (𝑅₁‘∅) ∈ 𝑦))
12		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑅₁‘𝑥) = (𝑅₁‘𝑤))
13	12	eleq1d 2819	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ (𝑅₁‘𝑤) ∈ 𝑦))
14		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑤 → (𝑅₁‘𝑥) = (𝑅₁‘suc 𝑤))
15	14	eleq1d 2819	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑤 → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ (𝑅₁‘suc 𝑤) ∈ 𝑦))
16		r10 9763	. . . . . . . . . . . . 13 ⊢ (𝑅₁‘∅) = ∅
17	16	eleq1i 2825	. . . . . . . . . . . 12 ⊢ ((𝑅₁‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
18	17	biimpri 227	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝑦 → (𝑅₁‘∅) ∈ 𝑦)
19	18	adantr 482	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁‘∅) ∈ 𝑦)
20		pweq 4617	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅₁‘𝑤) → 𝒫 𝑧 = 𝒫 (𝑅₁‘𝑤))
21	20	eleq1d 2819	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅₁‘𝑤) → (𝒫 𝑧 ∈ 𝑦 ↔ 𝒫 (𝑅₁‘𝑤) ∈ 𝑦))
22	21	rspccv 3610	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅₁‘𝑤) ∈ 𝑦 → 𝒫 (𝑅₁‘𝑤) ∈ 𝑦))
23		nnon 7861	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
24		r1suc 9765	. . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ω → ((𝑅₁‘suc 𝑤) ∈ 𝑦 ↔ 𝒫 (𝑅₁‘𝑤) ∈ 𝑦))
27	26	biimprcd 249	. . . . . . . . . . . . 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 492	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → ((𝑅₁‘𝑤) ∈ 𝑦 → (𝑅₁‘suc 𝑤) ∈ 𝑦)))
31	11, 13, 15, 19, 30	finds2 7891	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁‘𝑥) ∈ 𝑦))
32		eleq1 2822	. . . . . . . . . 10 ⊢ ((𝑅₁‘𝑥) = 𝑤 → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
33	32	biimpd 228	. . . . . . . . 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 3149	. . . . . . 7 ⊢ (∃𝑥 ∈ ω (𝑅₁‘𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → 𝑤 ∈ 𝑦))
36	9, 35	sylbi 216	. . . . . 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 3989	. . . 4 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁ “ ω) ⊆ 𝑦)
39		vex 3479	. . . . 5 ⊢ 𝑦 ∈ V
40	39	ssex 5322	. . . 4 ⊢ ((𝑅₁ “ ω) ⊆ 𝑦 → (𝑅₁ “ ω) ∈ V)
41	38, 40	syl 17	. . 3 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁ “ ω) ∈ V)
42		0ex 5308	. . . 4 ⊢ ∅ ∈ V
43		eleq1 2822	. . . . . 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 1925	. . . 4 ⊢ (𝑥 = ∅ → (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
46		axgroth6 10823	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))
47		simpr 486	. . . . . . . 8 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → 𝒫 𝑧 ∈ 𝑦)
48	47	ralimi 3084	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
49	48	anim2i 618	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦)) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
50	49	3adant3 1133	. . . . 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 1840	. . . 4 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
52	42, 45, 51	vtocl 3550	. . 3 ⊢ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
53	41, 52	exlimiiv 1935	. 2 ⊢ (𝑅₁ “ ω) ∈ V
54		f1dmex 7943	. 2 ⊢ (((𝑅₁ ↾ ω):ω–1-1→(𝑅₁ “ ω) ∧ (𝑅₁ “ ω) ∈ V) → ω ∈ V)
55	6, 53, 54	mp2an 691	1 ⊢ ω ∈ V

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 class class class wbr 5149 ↾ cres 5679 “ cima 5680 Oncon0 6365 suc csuc 6367 Fn wfn 6539 –1-1→wf1 6541 –1-1-onto→wf1o 6543 ‘cfv 6544 ωcom 7855 ≺ csdm 8938 𝑅₁cr1 9757

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-groth 10818

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-r1 9759

This theorem is referenced by: (None)

W3C validator