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Theorem grothac 9597
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9236). This can be put in a more conventional form via ween 8803 and dfac8 8902. 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). (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.
StepHypRef Expression
1 pweq 4138 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
21sseq1d 3616 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
31eleq1d 2688 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
42, 3anbi12d 746 . . . . . . . 8 (𝑥 = 𝑦 → ((𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ↔ (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢)))
54rspcva 3298 . . . . . . 7 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢))
65simpld 475 . . . . . 6 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → 𝒫 𝑦𝑢)
7 rabss 3663 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
87biimpri 218 . . . . . 6 (∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢) → {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢)
9 vex 3194 . . . . . . . . . 10 𝑦 ∈ V
109canth2 8058 . . . . . . . . 9 𝑦 ≺ 𝒫 𝑦
11 sdomdom 7928 . . . . . . . . 9 (𝑦 ≺ 𝒫 𝑦𝑦 ≼ 𝒫 𝑦)
1210, 11ax-mp 5 . . . . . . . 8 𝑦 ≼ 𝒫 𝑦
13 vex 3194 . . . . . . . . 9 𝑢 ∈ V
14 ssdomg 7946 . . . . . . . . 9 (𝑢 ∈ V → (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢))
1513, 14ax-mp 5 . . . . . . . 8 (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢)
16 domtr 7954 . . . . . . . 8 ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦𝑢) → 𝑦𝑢)
1712, 15, 16sylancr 694 . . . . . . 7 (𝒫 𝑦𝑢𝑦𝑢)
18 tskwe 8721 . . . . . . . 8 ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card)
1913, 18mpan 705 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢𝑢 ∈ dom card)
20 numdom 8806 . . . . . . . 8 ((𝑢 ∈ dom card ∧ 𝑦𝑢) → 𝑦 ∈ dom card)
2120expcom 451 . . . . . . 7 (𝑦𝑢 → (𝑢 ∈ dom card → 𝑦 ∈ dom card))
2217, 19, 21syl2im 40 . . . . . 6 (𝒫 𝑦𝑢 → ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢𝑦 ∈ dom card))
236, 8, 22syl2im 40 . . . . 5 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → (∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢) → 𝑦 ∈ dom card))
24233impia 1258 . . . 4 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢)) → 𝑦 ∈ dom card)
25 axgroth6 9595 . . . 4 𝑢(𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
2624, 25exlimiiv 1861 . . 3 𝑦 ∈ dom card
2726, 92th 254 . 2 (𝑦 ∈ dom card ↔ 𝑦 ∈ V)
2827eqriv 2623 1 dom card = V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  {crab 2916  Vcvv 3191  wss 3560  𝒫 cpw 4135   class class class wbr 4618  dom cdm 5079  cdom 7898  csdm 7899  cardccrd 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-groth 9590
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-wrecs 7353  df-recs 7414  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-card 8710
This theorem is referenced by:  axgroth3  9598
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