grothomex - Metamath Proof Explorer

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Theorem grothomex 10821

Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9635). 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 9767	. . . 4 ⊢ 𝑅₁:On–1-1→V
2		omsson 7856	. . . 4 ⊢ ω ⊆ On
3		f1ores 6845	. . . 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 6830	. . 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 9759	. . . . . . . 8 ⊢ 𝑅₁ Fn On
8		fvelimab 6962	. . . . . . . 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 6889	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑅₁‘𝑥) = (𝑅₁‘∅))
11	10	eleq1d 2819	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ (𝑅₁‘∅) ∈ 𝑦))
12		fveq2 6889	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑅₁‘𝑥) = (𝑅₁‘𝑤))
13	12	eleq1d 2819	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ (𝑅₁‘𝑤) ∈ 𝑦))
14		fveq2 6889	. . . . . . . . . . 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 9760	. . . . . . . . . . . . 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 4616	. . . . . . . . . . . . . . 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 7858	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
24		r1suc 9762	. . . . . . . . . . . . . . . 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 7888	. . . . . . . . 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 3988	. . . 4 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁ “ ω) ⊆ 𝑦)
39		vex 3479	. . . . 5 ⊢ 𝑦 ∈ V
40	39	ssex 5321	. . . 4 ⊢ ((𝑅₁ “ ω) ⊆ 𝑦 → (𝑅₁ “ ω) ∈ V)
41	38, 40	syl 17	. . 3 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁ “ ω) ∈ V)
42		0ex 5307	. . . 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 10820	. . . . 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 7940	. 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 3948 ∅c0 4322 𝒫 cpw 4602 class class class wbr 5148 ↾ cres 5678 “ cima 5679 Oncon0 6362 suc csuc 6364 Fn wfn 6536 –1-1→wf1 6538 –1-1-onto→wf1o 6540 ‘cfv 6541 ωcom 7852 ≺ csdm 8935 𝑅₁cr1 9754

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-groth 10815

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-r1 9756

This theorem is referenced by: (None)

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