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Theorem grudomon 10270
 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 5036 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
2 eleq1 2840 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
31, 2imbi12d 349 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝐵𝑥𝑈) ↔ (𝑦𝐵𝑦𝑈)))
43imbi2d 345 . . . . . 6 (𝑥 = 𝑦 → (((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵𝑥𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈))))
5 breq1 5036 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 eleq1 2840 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
75, 6imbi12d 349 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐵𝑥𝑈) ↔ (𝐴𝐵𝐴𝑈)))
87imbi2d 345 . . . . . 6 (𝑥 = 𝐴 → (((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵𝑥𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝐴𝐵𝐴𝑈))))
9 r19.21v 3107 . . . . . . 7 (∀𝑦𝑥 ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈)) ↔ ((𝑈 ∈ Univ ∧ 𝐵𝑈) → ∀𝑦𝑥 (𝑦𝐵𝑦𝑈)))
10 simpl1 1189 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → 𝑥 ∈ On)
11 vex 3414 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
12 onelss 6212 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1312imp 411 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
14 ssdomg 8574 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → (𝑦𝑥𝑦𝑥))
1511, 13, 14mpsyl 68 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
1610, 15sylan 584 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
17 simplr 769 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐵)
18 domtr 8581 . . . . . . . . . . . . . . 15 ((𝑦𝑥𝑥𝐵) → 𝑦𝐵)
1916, 17, 18syl2anc 588 . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → ∀𝑦𝑥 𝑦𝑈))
23 dfss3 3881 . . . . . . . . . . . . 13 (𝑥𝑈 ↔ ∀𝑦𝑥 𝑦𝑈)
24 domeng 8542 . . . . . . . . . . . . . . . 16 (𝐵𝑈 → (𝑥𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵)))
25243ad2ant3 1133 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵)))
2625biimpa 481 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → ∃𝑦(𝑥𝑦𝑦𝐵))
27 simpl2 1190 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → 𝑈 ∈ Univ)
28 gruss 10249 . . . . . . . . . . . . . . . . . . . . 21 ((𝑈 ∈ Univ ∧ 𝐵𝑈𝑦𝐵) → 𝑦𝑈)
29283expia 1119 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈))
30293adant1 1128 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈))
3130adantr 485 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (𝑦𝐵𝑦𝑈))
32 ensym 8577 . . . . . . . . . . . . . . . . . 18 (𝑥𝑦𝑦𝑥)
3331, 32anim12d1 613 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → ((𝑦𝐵𝑥𝑦) → (𝑦𝑈𝑦𝑥)))
3433ancomsd 470 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → ((𝑥𝑦𝑦𝐵) → (𝑦𝑈𝑦𝑥)))
3534eximdv 1919 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∃𝑦(𝑥𝑦𝑦𝐵) → ∃𝑦(𝑦𝑈𝑦𝑥)))
36 gruen 10265 . . . . . . . . . . . . . . . . . 18 ((𝑈 ∈ Univ ∧ 𝑥𝑈 ∧ (𝑦𝑈𝑦𝑥)) → 𝑥𝑈)
37363com23 1124 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ Univ ∧ (𝑦𝑈𝑦𝑥) ∧ 𝑥𝑈) → 𝑥𝑈)
38373exp 1117 . . . . . . . . . . . . . . . 16 (𝑈 ∈ Univ → ((𝑦𝑈𝑦𝑥) → (𝑥𝑈𝑥𝑈)))
3938exlimdv 1935 . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∀𝑦𝑥 𝑦𝑈𝑥𝑈))
4322, 42syld 47 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) ∧ 𝑥𝐵) → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → 𝑥𝑈))
4443ex 417 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵 → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → 𝑥𝑈)))
4544com23 86 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵𝑈) → (∀𝑦𝑥 (𝑦𝐵𝑦𝑈) → (𝑥𝐵𝑥𝑈)))
46453expib 1120 . . . . . . . 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 245 . . . . . 6 (𝑥 ∈ On → (∀𝑦𝑥 ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑦𝐵𝑦𝑈)) → ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝑥𝐵𝑥𝑈))))
494, 8, 48tfis3 7572 . . . . 5 (𝐴 ∈ On → ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝐴𝐵𝐴𝑈)))
5049com3l 89 . . . 4 ((𝑈 ∈ Univ ∧ 𝐵𝑈) → (𝐴𝐵 → (𝐴 ∈ On → 𝐴𝑈)))
5150impr 459 . . 3 ((𝑈 ∈ Univ ∧ (𝐵𝑈𝐴𝐵)) → (𝐴 ∈ On → 𝐴𝑈))
52513impia 1115 . 2 ((𝑈 ∈ Univ ∧ (𝐵𝑈𝐴𝐵) ∧ 𝐴 ∈ On) → 𝐴𝑈)
53523com23 1124 1 ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝐵𝑈𝐴𝐵)) → 𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∀wral 3071  Vcvv 3410   ⊆ wss 3859   class class class wbr 5033  Oncon0 6170   ≈ cen 8525   ≼ cdom 8526  Univcgru 10243 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6173  df-on 6174  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-er 8300  df-map 8419  df-en 8529  df-dom 8530  df-gru 10244 This theorem is referenced by:  gruina  10271  grur1  10273
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