MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smogt Structured version   Visualization version   GIF version

Theorem smogt 7861
Description: A strictly monotone ordinal function is greater than or equal to its argument. Exercise 1 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 23-Nov-2011.) (Revised by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
smogt ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝐶𝐴) → 𝐶 ⊆ (𝐹𝐶))

Proof of Theorem smogt
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . 6 (𝑥 = 𝐶𝑥 = 𝐶)
2 fveq2 6543 . . . . . 6 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
31, 2sseq12d 3925 . . . . 5 (𝑥 = 𝐶 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝐶 ⊆ (𝐹𝐶)))
43imbi2d 342 . . . 4 (𝑥 = 𝐶 → (((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹𝐶))))
5 smodm2 7849 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
653adant3 1125 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → Ord 𝐴)
7 simp3 1131 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥𝐴)
8 ordelord 6093 . . . . . . . . 9 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
96, 7, 8syl2anc 584 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → Ord 𝑥)
10 vex 3440 . . . . . . . . 9 𝑥 ∈ V
1110elon 6080 . . . . . . . 8 (𝑥 ∈ On ↔ Ord 𝑥)
129, 11sylibr 235 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ∈ On)
13 eleq1w 2865 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
14133anbi3d 1434 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ↔ (𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴)))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
16 fveq2 6543 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1715, 16sseq12d 3925 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑦 ⊆ (𝐹𝑦)))
1814, 17imbi12d 346 . . . . . . . 8 (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦))))
19 simpl1 1184 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → 𝐹 Fn 𝐴)
20 simpl2 1185 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → Smo 𝐹)
21 ordtr1 6114 . . . . . . . . . . . . . . 15 (Ord 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2221expcomd 417 . . . . . . . . . . . . . 14 (Ord 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
236, 7, 22sylc 65 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (𝑦𝑥𝑦𝐴))
2423imp 407 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → 𝑦𝐴)
25 pm2.27 42 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → 𝑦 ⊆ (𝐹𝑦)))
2619, 20, 24, 25syl3anc 1364 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → 𝑦 ⊆ (𝐹𝑦)))
2726ralimdva 3144 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → ∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦)))
2853adant3 1125 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝐴)
29 simp31 1202 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑥𝐴)
3028, 29, 8syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝑥)
31 simp32 1203 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦𝑥)
32 ordelord 6093 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
3330, 31, 32syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝑦)
34 smofvon2 7850 . . . . . . . . . . . . . . . . . . 19 (Smo 𝐹 → (𝐹𝑥) ∈ On)
35343ad2ant2 1127 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑥) ∈ On)
36 eloni 6081 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) ∈ On → Ord (𝐹𝑥))
3735, 36syl 17 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord (𝐹𝑥))
38 simp33 1204 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦 ⊆ (𝐹𝑦))
39 smoel2 7857 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥)) → (𝐹𝑦) ∈ (𝐹𝑥))
40393adantr3 1164 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑦) ∈ (𝐹𝑥))
41403impa 1103 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑦) ∈ (𝐹𝑥))
42 ordtr2 6115 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑦 ∧ Ord (𝐹𝑥)) → ((𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ (𝐹𝑥)) → 𝑦 ∈ (𝐹𝑥)))
4342imp 407 . . . . . . . . . . . . . . . . 17 (((Ord 𝑦 ∧ Ord (𝐹𝑥)) ∧ (𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ (𝐹𝑥))) → 𝑦 ∈ (𝐹𝑥))
4433, 37, 38, 41, 43syl22anc 835 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦 ∈ (𝐹𝑥))
45443expia 1114 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦)) → 𝑦 ∈ (𝐹𝑥)))
46453expd 1346 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥𝐴 → (𝑦𝑥 → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥)))))
47463impia 1110 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (𝑦𝑥 → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥))))
4847imp 407 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥)))
4948ralimdva 3144 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦) → ∀𝑦𝑥 𝑦 ∈ (𝐹𝑥)))
50 dfss3 3882 . . . . . . . . . . 11 (𝑥 ⊆ (𝐹𝑥) ↔ ∀𝑦𝑥 𝑦 ∈ (𝐹𝑥))
5149, 50syl6ibr 253 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦) → 𝑥 ⊆ (𝐹𝑥)))
5227, 51syldc 48 . . . . . . . . 9 (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)))
5352a1i 11 . . . . . . . 8 (𝑥 ∈ On → (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥))))
5418, 53tfis2 7432 . . . . . . 7 (𝑥 ∈ On → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)))
5512, 54mpcom 38 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥))
56553expia 1114 . . . . 5 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥𝐴𝑥 ⊆ (𝐹𝑥)))
5756com12 32 . . . 4 (𝑥𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹𝑥)))
584, 57vtoclga 3517 . . 3 (𝐶𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹𝐶)))
5958com12 32 . 2 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝐶𝐴𝐶 ⊆ (𝐹𝐶)))
60593impia 1110 1 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝐶𝐴) → 𝐶 ⊆ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  wss 3863  Ord word 6070  Oncon0 6071   Fn wfn 6225  cfv 6230  Smo wsmo 7839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-ord 6074  df-on 6075  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-fv 6238  df-smo 7840
This theorem is referenced by:  smorndom  7862  oismo  8855
  Copyright terms: Public domain W3C validator