Theorem grothac 10810

Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10448). This can be put in a more conventional form via ween 10015 and dfac8 10115. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see https://fomarchive.ugent.be/2008-March/012783.html 10115). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
grothac	⊢ dom card = V

Proof of Theorem grothac

Dummy variables 𝑥 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4576	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
2	1	sseq1d 3968	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ⊆ 𝑢 ↔ 𝒫 𝑦 ⊆ 𝑢))
3	1	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ∈ 𝑢 ↔ 𝒫 𝑦 ∈ 𝑢))
4	2, 3	anbi12d 643	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ↔ (𝒫 𝑦 ⊆ 𝑢 ∧ 𝒫 𝑦 ∈ 𝑢)))
5	4	rspcva 3579	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → (𝒫 𝑦 ⊆ 𝑢 ∧ 𝒫 𝑦 ∈ 𝑢))
6	5	simpld 499	. . . . . 6 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → 𝒫 𝑦 ⊆ 𝑢)
7		rabss 4024	. . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢))
8	7	biimpri 231	. . . . . 6 ⊢ (∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢) → {𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢)
9		vex 3459	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9	canth2 9114	. . . . . . . . 9 ⊢ 𝑦 ≺ 𝒫 𝑦
11		sdomdom 8973	. . . . . . . . 9 ⊢ (𝑦 ≺ 𝒫 𝑦 → 𝑦 ≼ 𝒫 𝑦)
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝑦 ≼ 𝒫 𝑦
13		ssdomg 8993	. . . . . . . . 9 ⊢ (𝑢 ∈ V → (𝒫 𝑦 ⊆ 𝑢 → 𝒫 𝑦 ≼ 𝑢))
14	13	elv 3460	. . . . . . . 8 ⊢ (𝒫 𝑦 ⊆ 𝑢 → 𝒫 𝑦 ≼ 𝑢)
15		domtr 9000	. . . . . . . 8 ⊢ ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦 ≼ 𝑢) → 𝑦 ≼ 𝑢)
16	12, 14, 15	sylancr 598	. . . . . . 7 ⊢ (𝒫 𝑦 ⊆ 𝑢 → 𝑦 ≼ 𝑢)
17		vex 3459	. . . . . . . 8 ⊢ 𝑢 ∈ V
18		tskwe 9932	. . . . . . . 8 ⊢ ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card)
19	17, 18	mpan 702	. . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 → 𝑢 ∈ dom card)
20		numdom 10018	. . . . . . . 8 ⊢ ((𝑢 ∈ dom card ∧ 𝑦 ≼ 𝑢) → 𝑦 ∈ dom card)
21	20	expcom 418	. . . . . . 7 ⊢ (𝑦 ≼ 𝑢 → (𝑢 ∈ dom card → 𝑦 ∈ dom card))
22	16, 19, 21	syl2im 41	. . . . . 6 ⊢ (𝒫 𝑦 ⊆ 𝑢 → ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 → 𝑦 ∈ dom card))
23	6, 8, 22	syl2im 41	. . . . 5 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → (∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢) → 𝑦 ∈ dom card))
24	23	3impia 1135	. . . 4 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢)) → 𝑦 ∈ dom card)
25		axgroth6 10808	. . . 4 ⊢ ∃𝑢(𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢))
26	24, 25	exlimiiv 1961	. . 3 ⊢ 𝑦 ∈ dom card
27	26, 9	2th 267	. 2 ⊢ (𝑦 ∈ dom card ↔ 𝑦 ∈ V)
28	27	eqriv 2760	1 ⊢ dom card = V

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562   class class class wbr 5109  dom cdm 5661   ≼ cdom 8937   ≺ csdm 8938  cardccrd 9917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-groth 10803

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-card 9921

This theorem is referenced by:  axgroth3  10811