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Theorem smogt 7703
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 6411 . . . . . 6 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
31, 2sseq12d 3830 . . . . 5 (𝑥 = 𝐶 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝐶 ⊆ (𝐹𝐶)))
43imbi2d 332 . . . 4 (𝑥 = 𝐶 → (((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹𝐶))))
5 smodm2 7691 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
653adant3 1163 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → Ord 𝐴)
7 simp3 1169 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥𝐴)
8 ordelord 5963 . . . . . . . . 9 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
96, 7, 8syl2anc 580 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → Ord 𝑥)
10 vex 3388 . . . . . . . . 9 𝑥 ∈ V
1110elon 5950 . . . . . . . 8 (𝑥 ∈ On ↔ Ord 𝑥)
129, 11sylibr 226 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ∈ On)
13 eleq1w 2861 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
14133anbi3d 1567 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ↔ (𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴)))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
16 fveq2 6411 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1715, 16sseq12d 3830 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑦 ⊆ (𝐹𝑦)))
1814, 17imbi12d 336 . . . . . . . 8 (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦))))
19 simpl1 1243 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → 𝐹 Fn 𝐴)
20 simpl2 1245 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → Smo 𝐹)
21 ordtr1 5984 . . . . . . . . . . . . . . 15 (Ord 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2221expcomd 407 . . . . . . . . . . . . . 14 (Ord 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
236, 7, 22sylc 65 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (𝑦𝑥𝑦𝐴))
2423imp 396 . . . . . . . . . . . 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 1491 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → 𝑦 ⊆ (𝐹𝑦)))
2726ralimdva 3143 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → ∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦)))
2853adant3 1163 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝐴)
29 simp31 1267 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑥𝐴)
3028, 29, 8syl2anc 580 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝑥)
31 simp32 1268 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦𝑥)
32 ordelord 5963 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
3330, 31, 32syl2anc 580 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝑦)
34 smofvon2 7692 . . . . . . . . . . . . . . . . . . 19 (Smo 𝐹 → (𝐹𝑥) ∈ On)
35343ad2ant2 1165 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑥) ∈ On)
36 eloni 5951 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) ∈ On → Ord (𝐹𝑥))
3735, 36syl 17 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord (𝐹𝑥))
38 simp33 1269 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦 ⊆ (𝐹𝑦))
39 smoel2 7699 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥)) → (𝐹𝑦) ∈ (𝐹𝑥))
40393adantr3 1213 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑦) ∈ (𝐹𝑥))
41403impa 1137 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑦) ∈ (𝐹𝑥))
42 ordtr2 5985 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑦 ∧ Ord (𝐹𝑥)) → ((𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ (𝐹𝑥)) → 𝑦 ∈ (𝐹𝑥)))
4342imp 396 . . . . . . . . . . . . . . . . 17 (((Ord 𝑦 ∧ Ord (𝐹𝑥)) ∧ (𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ (𝐹𝑥))) → 𝑦 ∈ (𝐹𝑥))
4433, 37, 38, 41, 43syl22anc 868 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦 ∈ (𝐹𝑥))
45443expia 1151 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦)) → 𝑦 ∈ (𝐹𝑥)))
46453expd 1463 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥𝐴 → (𝑦𝑥 → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥)))))
47463impia 1146 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (𝑦𝑥 → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥))))
4847imp 396 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥)))
4948ralimdva 3143 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦) → ∀𝑦𝑥 𝑦 ∈ (𝐹𝑥)))
50 dfss3 3787 . . . . . . . . . . 11 (𝑥 ⊆ (𝐹𝑥) ↔ ∀𝑦𝑥 𝑦 ∈ (𝐹𝑥))
5149, 50syl6ibr 244 . . . . . . . . . 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 7290 . . . . . . 7 (𝑥 ∈ On → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)))
5512, 54mpcom 38 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥))
56553expia 1151 . . . . 5 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥𝐴𝑥 ⊆ (𝐹𝑥)))
5756com12 32 . . . 4 (𝑥𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹𝑥)))
584, 57vtoclga 3460 . . 3 (𝐶𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹𝐶)))
5958com12 32 . 2 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝐶𝐴𝐶 ⊆ (𝐹𝐶)))
60593impia 1146 1 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝐶𝐴) → 𝐶 ⊆ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  wss 3769  Ord word 5940  Oncon0 5941   Fn wfn 6096  cfv 6101  Smo wsmo 7681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-ord 5944  df-on 5945  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-smo 7682
This theorem is referenced by:  smorndom  7704  oismo  8687
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