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Theorem grudomon 10231
Description: Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
grudomon ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝐵𝑈𝐴𝐵)) → 𝐴𝑈)

Proof of Theorem grudomon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5065 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
2 eleq1 2904 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
31, 2imbi12d 346 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝐵𝑥𝑈) ↔ (𝑦𝐵𝑦𝑈)))
43imbi2d 342 . . . . . 6 (𝑥 = 𝑦 → (((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵𝑥𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈))))
5 breq1 5065 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 eleq1 2904 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
75, 6imbi12d 346 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐵𝑥𝑈) ↔ (𝐴𝐵𝐴𝑈)))
87imbi2d 342 . . . . . 6 (𝑥 = 𝐴 → (((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵𝑥𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝐴𝐵𝐴𝑈))))
9 r19.21v 3179 . . . . . . 7 (∀𝑦𝑥 ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵𝑈) → ∀𝑦𝑥 (𝑦𝐵𝑦𝑈)))
10 simpl1 1185 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → 𝑥 ∈ On)
11 vex 3502 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
12 onelss 6230 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1312imp 407 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
14 ssdomg 8547 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → (𝑦𝑥𝑦𝑥))
1511, 13, 14mpsyl 68 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
1610, 15sylan 580 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
17 simplr 765 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐵)
18 domtr 8554 . . . . . . . . . . . . . . 15 ((𝑦𝑥𝑥𝐵) → 𝑦𝐵)
1916, 17, 18syl2anc 584 . . . . . . . . . . . . . 14 ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝐵)
20 pm2.27 42 . . . . . . . . . . . . . 14 (𝑦𝐵 → ((𝑦𝐵𝑦𝑈) → 𝑦𝑈))
2119, 20syl 17 . . . . . . . . . . . . 13 ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) ∧ 𝑦𝑥) → ((𝑦𝐵𝑦𝑈) → 𝑦𝑈))
2221ralimdva 3181 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → ∀𝑦𝑥 𝑦𝑈))
23 dfss3 3959 . . . . . . . . . . . . 13 (𝑥𝑈 ↔ ∀𝑦𝑥 𝑦𝑈)
24 domeng 8515 . . . . . . . . . . . . . . . 16 (𝐵𝑈 → (𝑥𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵)))
25243ad2ant3 1129 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵)))
2625biimpa 477 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → ∃𝑦(𝑥𝑦𝑦𝐵))
27 simpl2 1186 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → 𝑈 ∈ Univ)
28 gruss 10210 . . . . . . . . . . . . . . . . . . . . 21 ((𝑈 ∈ Univ ∧ 𝐵𝑈𝑦𝐵) → 𝑦𝑈)
29283expia 1115 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈))
30293adant1 1124 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈))
3130adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (𝑦𝐵𝑦𝑈))
32 ensym 8550 . . . . . . . . . . . . . . . . . 18 (𝑥𝑦𝑦𝑥)
3331, 32anim12d1 609 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → ((𝑦𝐵𝑥𝑦) → (𝑦𝑈𝑦𝑥)))
3433ancomsd 466 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → ((𝑥𝑦𝑦𝐵) → (𝑦𝑈𝑦𝑥)))
3534eximdv 1911 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∃𝑦(𝑥𝑦𝑦𝐵) → ∃𝑦(𝑦𝑈𝑦𝑥)))
36 gruen 10226 . . . . . . . . . . . . . . . . . 18 ((𝑈 ∈ Univ ∧ 𝑥𝑈 ∧ (𝑦𝑈𝑦𝑥)) → 𝑥𝑈)
37363com23 1120 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ Univ ∧ (𝑦𝑈𝑦𝑥) ∧ 𝑥𝑈) → 𝑥𝑈)
38373exp 1113 . . . . . . . . . . . . . . . 16 (𝑈 ∈ Univ → ((𝑦𝑈𝑦𝑥) → (𝑥𝑈𝑥𝑈)))
3938exlimdv 1927 . . . . . . . . . . . . . . 15 (𝑈 ∈ Univ → (∃𝑦(𝑦𝑈𝑦𝑥) → (𝑥𝑈𝑥𝑈)))
4027, 35, 39sylsyld 61 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∃𝑦(𝑥𝑦𝑦𝐵) → (𝑥𝑈𝑥𝑈)))
4126, 40mpd 15 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (𝑥𝑈𝑥𝑈))
4223, 41syl5bir 244 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∀𝑦𝑥 𝑦𝑈𝑥𝑈))
4322, 42syld 47 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → 𝑥𝑈))
4443ex 413 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵 → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → 𝑥𝑈)))
4544com23 86 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → (𝑥𝐵𝑥𝑈)))
46453expib 1116 . . . . . . . 8 (𝑥 ∈ On → ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → (𝑥𝐵𝑥𝑈))))
4746a2d 29 . . . . . . 7 (𝑥 ∈ On → (((𝑈 ∈ Univ ∧ 𝐵𝑈) → ∀𝑦𝑥 (𝑦𝐵𝑦𝑈)) → ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵𝑥𝑈))))
489, 47syl5bi 243 . . . . . 6 (𝑥 ∈ On → (∀𝑦𝑥 ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈)) → ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵𝑥𝑈))))
494, 8, 48tfis3 7563 . . . . 5 (𝐴 ∈ On → ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝐴𝐵𝐴𝑈)))
5049com3l 89 . . . 4 ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝐴𝐵 → (𝐴 ∈ On → 𝐴𝑈)))
5150impr 455 . . 3 ((𝑈 ∈ Univ ∧ (𝐵𝑈𝐴𝐵)) → (𝐴 ∈ On → 𝐴𝑈))
52513impia 1111 . 2 ((𝑈 ∈ Univ ∧ (𝐵𝑈𝐴𝐵) ∧ 𝐴 ∈ On) → 𝐴𝑈)
53523com23 1120 1 ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝐵𝑈𝐴𝐵)) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2106  wral 3142  Vcvv 3499  wss 3939   class class class wbr 5062  Oncon0 6188  cen 8498  cdom 8499  Univcgru 10204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-gru 10205
This theorem is referenced by:  gruina  10232  grur1  10234
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