grothomex - Metamath Proof Explorer

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Theorem grothomex 10826

Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9640). 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 9772	. . . 4 ⊢ 𝑅₁:On–1-1→V
2		omsson 7861	. . . 4 ⊢ ω ⊆ On
3		f1ores 6846	. . . 4 ⊢ ((𝑅₁:On–1-1→V ∧ ω ⊆ On) → (𝑅₁ ↾ ω):ω–1-1-onto→(𝑅₁ “ ω))
4	1, 2, 3	mp2an 688	. . 3 ⊢ (𝑅₁ ↾ ω):ω–1-1-onto→(𝑅₁ “ ω)
5		f1of1 6831	. . 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 9764	. . . . . . . 8 ⊢ 𝑅₁ Fn On
8		fvelimab 6963	. . . . . . . 8 ⊢ ((𝑅₁ Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅₁ “ ω) ↔ ∃𝑥 ∈ ω (𝑅₁‘𝑥) = 𝑤))
9	7, 2, 8	mp2an 688	. . . . . . 7 ⊢ (𝑤 ∈ (𝑅₁ “ ω) ↔ ∃𝑥 ∈ ω (𝑅₁‘𝑥) = 𝑤)
10		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑅₁‘𝑥) = (𝑅₁‘∅))
11	10	eleq1d 2816	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ (𝑅₁‘∅) ∈ 𝑦))
12		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑅₁‘𝑥) = (𝑅₁‘𝑤))
13	12	eleq1d 2816	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ (𝑅₁‘𝑤) ∈ 𝑦))
14		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑤 → (𝑅₁‘𝑥) = (𝑅₁‘suc 𝑤))
15	14	eleq1d 2816	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑤 → ((𝑅₁‘𝑥) ∈ 𝑦 ↔ (𝑅₁‘suc 𝑤) ∈ 𝑦))
16		r10 9765	. . . . . . . . . . . . 13 ⊢ (𝑅₁‘∅) = ∅
17	16	eleq1i 2822	. . . . . . . . . . . 12 ⊢ ((𝑅₁‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
18	17	biimpri 227	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝑦 → (𝑅₁‘∅) ∈ 𝑦)
19	18	adantr 479	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁‘∅) ∈ 𝑦)
20		pweq 4615	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅₁‘𝑤) → 𝒫 𝑧 = 𝒫 (𝑅₁‘𝑤))
21	20	eleq1d 2816	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅₁‘𝑤) → (𝒫 𝑧 ∈ 𝑦 ↔ 𝒫 (𝑅₁‘𝑤) ∈ 𝑦))
22	21	rspccv 3608	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 → ((𝑅₁‘𝑤) ∈ 𝑦 → 𝒫 (𝑅₁‘𝑤) ∈ 𝑦))
23		nnon 7863	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → 𝑤 ∈ On)
24		r1suc 9767	. . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . 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 489	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → ((𝑅₁‘𝑤) ∈ 𝑦 → (𝑅₁‘suc 𝑤) ∈ 𝑦)))
31	11, 13, 15, 19, 30	finds2 7893	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁‘𝑥) ∈ 𝑦))
32		eleq1 2819	. . . . . . . . . 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 3146	. . . . . . 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 3987	. . . 4 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁ “ ω) ⊆ 𝑦)
39		vex 3476	. . . . 5 ⊢ 𝑦 ∈ V
40	39	ssex 5320	. . . 4 ⊢ ((𝑅₁ “ ω) ⊆ 𝑦 → (𝑅₁ “ ω) ∈ V)
41	38, 40	syl 17	. . 3 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) → (𝑅₁ “ ω) ∈ V)
42		0ex 5306	. . . 4 ⊢ ∅ ∈ V
43		eleq1 2819	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
44	43	anbi1d 628	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
45	44	exbidv 1922	. . . 4 ⊢ (𝑥 = ∅ → (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)))
46		axgroth6 10825	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))
47		simpr 483	. . . . . . . 8 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → 𝒫 𝑧 ∈ 𝑦)
48	47	ralimi 3081	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
49	48	anim2i 615	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦)) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
50	49	3adant3 1130	. . . . 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 3544	. . 3 ⊢ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦)
53	41, 52	exlimiiv 1932	. 2 ⊢ (𝑅₁ “ ω) ∈ V
54		f1dmex 7945	. 2 ⊢ (((𝑅₁ ↾ ω):ω–1-1→(𝑅₁ “ ω) ∧ (𝑅₁ “ ω) ∈ V) → ω ∈ V)
55	6, 53, 54	mp2an 688	1 ⊢ ω ∈ V

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 class class class wbr 5147 ↾ cres 5677 “ cima 5678 Oncon0 6363 suc csuc 6365 Fn wfn 6537 –1-1→wf1 6539 –1-1-onto→wf1o 6541 ‘cfv 6542 ωcom 7857 ≺ csdm 8940 𝑅₁cr1 9759

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-groth 10820

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-r1 9761

This theorem is referenced by: (None)

W3C validator