Theorem grothac 10871

Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10510). This can be put in a more conventional form via ween 10076 and dfac8 10177. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10177). (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 4613	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
2	1	sseq1d 4014	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ⊆ 𝑢 ↔ 𝒫 𝑦 ⊆ 𝑢))
3	1	eleq1d 2825	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ∈ 𝑢 ↔ 𝒫 𝑦 ∈ 𝑢))
4	2, 3	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ↔ (𝒫 𝑦 ⊆ 𝑢 ∧ 𝒫 𝑦 ∈ 𝑢)))
5	4	rspcva 3619	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → (𝒫 𝑦 ⊆ 𝑢 ∧ 𝒫 𝑦 ∈ 𝑢))
6	5	simpld 494	. . . . . 6 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → 𝒫 𝑦 ⊆ 𝑢)
7		rabss 4071	. . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢))
8	7	biimpri 228	. . . . . 6 ⊢ (∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢) → {𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢)
9		vex 3483	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9	canth2 9171	. . . . . . . . 9 ⊢ 𝑦 ≺ 𝒫 𝑦
11		sdomdom 9021	. . . . . . . . 9 ⊢ (𝑦 ≺ 𝒫 𝑦 → 𝑦 ≼ 𝒫 𝑦)
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝑦 ≼ 𝒫 𝑦
13		ssdomg 9041	. . . . . . . . 9 ⊢ (𝑢 ∈ V → (𝒫 𝑦 ⊆ 𝑢 → 𝒫 𝑦 ≼ 𝑢))
14	13	elv 3484	. . . . . . . 8 ⊢ (𝒫 𝑦 ⊆ 𝑢 → 𝒫 𝑦 ≼ 𝑢)
15		domtr 9048	. . . . . . . 8 ⊢ ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦 ≼ 𝑢) → 𝑦 ≼ 𝑢)
16	12, 14, 15	sylancr 587	. . . . . . 7 ⊢ (𝒫 𝑦 ⊆ 𝑢 → 𝑦 ≼ 𝑢)
17		vex 3483	. . . . . . . 8 ⊢ 𝑢 ∈ V
18		tskwe 9991	. . . . . . . 8 ⊢ ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card)
19	17, 18	mpan 690	. . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 → 𝑢 ∈ dom card)
20		numdom 10079	. . . . . . . 8 ⊢ ((𝑢 ∈ dom card ∧ 𝑦 ≼ 𝑢) → 𝑦 ∈ dom card)
21	20	expcom 413	. . . . . . 7 ⊢ (𝑦 ≼ 𝑢 → (𝑢 ∈ dom card → 𝑦 ∈ dom card))
22	16, 19, 21	syl2im 40	. . . . . 6 ⊢ (𝒫 𝑦 ⊆ 𝑢 → ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 → 𝑦 ∈ dom card))
23	6, 8, 22	syl2im 40	. . . . 5 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → (∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢) → 𝑦 ∈ dom card))
24	23	3impia 1117	. . . 4 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢)) → 𝑦 ∈ dom card)
25		axgroth6 10869	. . . 4 ⊢ ∃𝑢(𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢))
26	24, 25	exlimiiv 1930	. . 3 ⊢ 𝑦 ∈ dom card
27	26, 9	2th 264	. 2 ⊢ (𝑦 ∈ dom card ↔ 𝑦 ∈ V)
28	27	eqriv 2733	1 ⊢ dom card = V

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  {crab 3435  Vcvv 3479   ⊆ wss 3950  𝒫 cpw 4599   class class class wbr 5142  dom cdm 5684   ≼ cdom 8984   ≺ csdm 8985  cardccrd 9976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-groth 10864

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-card 9980

This theorem is referenced by:  axgroth3  10872