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Theorem smogt 8226
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 6801 . . . . . 6 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
31, 2sseq12d 3960 . . . . 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 8214 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
653adant3 1132 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → Ord 𝐴)
7 simp3 1138 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥𝐴)
8 ordelord 6300 . . . . . . . . 9 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
96, 7, 8syl2anc 585 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → Ord 𝑥)
10 vex 3442 . . . . . . . . 9 𝑥 ∈ V
1110elon 6287 . . . . . . . 8 (𝑥 ∈ On ↔ Ord 𝑥)
129, 11sylibr 234 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ∈ On)
13 eleq1w 2819 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
14133anbi3d 1442 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ↔ (𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴)))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
16 fveq2 6801 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1715, 16sseq12d 3960 . . . . . . . . 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 1191 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → 𝐹 Fn 𝐴)
20 simpl2 1192 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → Smo 𝐹)
21 ordtr1 6321 . . . . . . . . . . . . . . 15 (Ord 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2221expcomd 418 . . . . . . . . . . . . . 14 (Ord 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
236, 7, 22sylc 65 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (𝑦𝑥𝑦𝐴))
2423imp 408 . . . . . . . . . . . 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 1371 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → 𝑦 ⊆ (𝐹𝑦)))
2726ralimdva 3161 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → ∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦)))
2853adant3 1132 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝐴)
29 simp31 1209 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑥𝐴)
3028, 29, 8syl2anc 585 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝑥)
31 simp32 1210 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦𝑥)
32 ordelord 6300 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
3330, 31, 32syl2anc 585 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝑦)
34 smofvon2 8215 . . . . . . . . . . . . . . . . . . 19 (Smo 𝐹 → (𝐹𝑥) ∈ On)
35343ad2ant2 1134 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑥) ∈ On)
36 eloni 6288 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) ∈ On → Ord (𝐹𝑥))
3735, 36syl 17 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord (𝐹𝑥))
38 simp33 1211 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦 ⊆ (𝐹𝑦))
39 smoel2 8222 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥)) → (𝐹𝑦) ∈ (𝐹𝑥))
40393adantr3 1171 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑦) ∈ (𝐹𝑥))
41403impa 1110 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑦) ∈ (𝐹𝑥))
42 ordtr2 6322 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑦 ∧ Ord (𝐹𝑥)) → ((𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ (𝐹𝑥)) → 𝑦 ∈ (𝐹𝑥)))
4342imp 408 . . . . . . . . . . . . . . . . 17 (((Ord 𝑦 ∧ Ord (𝐹𝑥)) ∧ (𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ (𝐹𝑥))) → 𝑦 ∈ (𝐹𝑥))
4433, 37, 38, 41, 43syl22anc 837 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦 ∈ (𝐹𝑥))
45443expia 1121 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦)) → 𝑦 ∈ (𝐹𝑥)))
46453expd 1353 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥𝐴 → (𝑦𝑥 → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥)))))
47463impia 1117 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (𝑦𝑥 → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥))))
4847imp 408 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥)))
4948ralimdva 3161 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦) → ∀𝑦𝑥 𝑦 ∈ (𝐹𝑥)))
50 dfss3 3915 . . . . . . . . . . 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 7732 . . . . . . 7 (𝑥 ∈ On → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)))
5512, 54mpcom 38 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥))
56553expia 1121 . . . . 5 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥𝐴𝑥 ⊆ (𝐹𝑥)))
5756com12 32 . . . 4 (𝑥𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹𝑥)))
584, 57vtoclga 3519 . . 3 (𝐶𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹𝐶)))
5958com12 32 . 2 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝐶𝐴𝐶 ⊆ (𝐹𝐶)))
60593impia 1117 1 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝐶𝐴) → 𝐶 ⊆ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1087   = wceq 1539  wcel 2104  wral 3062  wss 3893  Ord word 6277  Oncon0 6278   Fn wfn 6450  cfv 6455  Smo wsmo 8204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5233  ax-nul 5240  ax-pr 5362
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3280  df-v 3440  df-dif 3896  df-un 3898  df-in 3900  df-ss 3910  df-pss 3912  df-nul 4264  df-if 4467  df-pw 4542  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4846  df-br 5083  df-opab 5145  df-tr 5200  df-id 5497  df-eprel 5503  df-po 5511  df-so 5512  df-fr 5552  df-we 5554  df-xp 5603  df-rel 5604  df-cnv 5605  df-co 5606  df-dm 5607  df-rn 5608  df-ord 6281  df-on 6282  df-iota 6407  df-fun 6457  df-fn 6458  df-f 6459  df-fv 6463  df-smo 8205
This theorem is referenced by:  smorndom  8227  oismo  9340
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