Theorem grothac 10825

Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10464). This can be put in a more conventional form via ween 10030 and dfac8 10130. 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 10130). (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 4617	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
2	1	sseq1d 4014	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ⊆ 𝑢 ↔ 𝒫 𝑦 ⊆ 𝑢))
3	1	eleq1d 2819	. . . . . . . . 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 3611	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → (𝒫 𝑦 ⊆ 𝑢 ∧ 𝒫 𝑦 ∈ 𝑢))
6	5	simpld 496	. . . . . 6 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → 𝒫 𝑦 ⊆ 𝑢)
7		rabss 4070	. . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢))
8	7	biimpri 227	. . . . . 6 ⊢ (∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢) → {𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢)
9		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9	canth2 9130	. . . . . . . . 9 ⊢ 𝑦 ≺ 𝒫 𝑦
11		sdomdom 8976	. . . . . . . . 9 ⊢ (𝑦 ≺ 𝒫 𝑦 → 𝑦 ≼ 𝒫 𝑦)
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝑦 ≼ 𝒫 𝑦
13		ssdomg 8996	. . . . . . . . 9 ⊢ (𝑢 ∈ V → (𝒫 𝑦 ⊆ 𝑢 → 𝒫 𝑦 ≼ 𝑢))
14	13	elv 3481	. . . . . . . 8 ⊢ (𝒫 𝑦 ⊆ 𝑢 → 𝒫 𝑦 ≼ 𝑢)
15		domtr 9003	. . . . . . . 8 ⊢ ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦 ≼ 𝑢) → 𝑦 ≼ 𝑢)
16	12, 14, 15	sylancr 588	. . . . . . 7 ⊢ (𝒫 𝑦 ⊆ 𝑢 → 𝑦 ≼ 𝑢)
17		vex 3479	. . . . . . . 8 ⊢ 𝑢 ∈ V
18		tskwe 9945	. . . . . . . 8 ⊢ ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card)
19	17, 18	mpan 689	. . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 → 𝑢 ∈ dom card)
20		numdom 10033	. . . . . . . 8 ⊢ ((𝑢 ∈ dom card ∧ 𝑦 ≼ 𝑢) → 𝑦 ∈ dom card)
21	20	expcom 415	. . . . . . 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 1118	. . . 4 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢)) → 𝑦 ∈ dom card)
25		axgroth6 10823	. . . 4 ⊢ ∃𝑢(𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢))
26	24, 25	exlimiiv 1935	. . 3 ⊢ 𝑦 ∈ dom card
27	26, 9	2th 264	. 2 ⊢ (𝑦 ∈ dom card ↔ 𝑦 ∈ V)
28	27	eqriv 2730	1 ⊢ dom card = V

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475   ⊆ wss 3949  𝒫 cpw 4603   class class class wbr 5149  dom cdm 5677   ≼ cdom 8937   ≺ csdm 8938  cardccrd 9930

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-rmo 3377  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-int 4952  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-se 5633  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-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-isom 6553  df-riota 7365  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-card 9934

This theorem is referenced by:  axgroth3  10826