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Theorem smogt 7325
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 6085 . . . . . 6 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
31, 2sseq12d 3593 . . . . 5 (𝑥 = 𝐶 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝐶 ⊆ (𝐹𝐶)))
43imbi2d 328 . . . 4 (𝑥 = 𝐶 → (((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹𝐶))))
5 smodm2 7313 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
653adant3 1073 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → Ord 𝐴)
7 simp3 1055 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥𝐴)
8 ordelord 5645 . . . . . . . . 9 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
96, 7, 8syl2anc 690 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → Ord 𝑥)
10 vex 3172 . . . . . . . . 9 𝑥 ∈ V
1110elon 5632 . . . . . . . 8 (𝑥 ∈ On ↔ Ord 𝑥)
129, 11sylibr 222 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ∈ On)
13 eleq1 2672 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
14133anbi3d 1396 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ↔ (𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴)))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
16 fveq2 6085 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1715, 16sseq12d 3593 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ⊆ (𝐹𝑥) ↔ 𝑦 ⊆ (𝐹𝑦)))
1814, 17imbi12d 332 . . . . . . . 8 (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦))))
19 simpl1 1056 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → 𝐹 Fn 𝐴)
20 simpl2 1057 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → Smo 𝐹)
21 ordtr1 5667 . . . . . . . . . . . . . . . 16 (Ord 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2221expcomd 452 . . . . . . . . . . . . . . 15 (Ord 𝐴 → (𝑥𝐴 → (𝑦𝑥𝑦𝐴)))
236, 7, 22sylc 62 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (𝑦𝑥𝑦𝐴))
2423imp 443 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → 𝑦𝐴)
25 pm2.27 40 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → 𝑦 ⊆ (𝐹𝑦)))
2619, 20, 24, 25syl3anc 1317 . . . . . . . . . . . 12 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → 𝑦 ⊆ (𝐹𝑦)))
2726ralimdva 2941 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → ∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦)))
2853adant3 1073 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝐴)
29 simp31 1089 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑥𝐴)
3028, 29, 8syl2anc 690 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝑥)
31 simp32 1090 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦𝑥)
32 ordelord 5645 . . . . . . . . . . . . . . . . . . 19 ((Ord 𝑥𝑦𝑥) → Ord 𝑦)
3330, 31, 32syl2anc 690 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord 𝑦)
34 smofvon2 7314 . . . . . . . . . . . . . . . . . . . 20 (Smo 𝐹 → (𝐹𝑥) ∈ On)
35343ad2ant2 1075 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑥) ∈ On)
36 eloni 5633 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑥) ∈ On → Ord (𝐹𝑥))
3735, 36syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → Ord (𝐹𝑥))
38 simp33 1091 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦 ⊆ (𝐹𝑦))
39 smoel2 7321 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥)) → (𝐹𝑦) ∈ (𝐹𝑥))
40393adantr3 1214 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑦) ∈ (𝐹𝑥))
41403impa 1250 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → (𝐹𝑦) ∈ (𝐹𝑥))
42 ordtr2 5668 . . . . . . . . . . . . . . . . . . 19 ((Ord 𝑦 ∧ Ord (𝐹𝑥)) → ((𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ (𝐹𝑥)) → 𝑦 ∈ (𝐹𝑥)))
4342imp 443 . . . . . . . . . . . . . . . . . 18 (((Ord 𝑦 ∧ Ord (𝐹𝑥)) ∧ (𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ (𝐹𝑥))) → 𝑦 ∈ (𝐹𝑥))
4433, 37, 38, 41, 43syl22anc 1318 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦))) → 𝑦 ∈ (𝐹𝑥))
45443expia 1258 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑥𝐴𝑦𝑥𝑦 ⊆ (𝐹𝑦)) → 𝑦 ∈ (𝐹𝑥)))
46453expd 1275 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥𝐴 → (𝑦𝑥 → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥)))))
47463impia 1252 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (𝑦𝑥 → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥))))
4847imp 443 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) ∧ 𝑦𝑥) → (𝑦 ⊆ (𝐹𝑦) → 𝑦 ∈ (𝐹𝑥)))
4948ralimdva 2941 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦) → ∀𝑦𝑥 𝑦 ∈ (𝐹𝑥)))
50 dfss3 3554 . . . . . . . . . . . 12 (𝑥 ⊆ (𝐹𝑥) ↔ ∀𝑦𝑥 𝑦 ∈ (𝐹𝑥))
5149, 50syl6ibr 240 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 𝑦 ⊆ (𝐹𝑦) → 𝑥 ⊆ (𝐹𝑥)))
5227, 51syld 45 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → 𝑥 ⊆ (𝐹𝑥)))
5352com12 32 . . . . . . . . 9 (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)))
5453a1i 11 . . . . . . . 8 (𝑥 ∈ On → (∀𝑦𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑦𝐴) → 𝑦 ⊆ (𝐹𝑦)) → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥))))
5518, 54tfis2 6922 . . . . . . 7 (𝑥 ∈ On → ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥)))
5612, 55mpcom 37 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝑥𝐴) → 𝑥 ⊆ (𝐹𝑥))
57563expia 1258 . . . . 5 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥𝐴𝑥 ⊆ (𝐹𝑥)))
5857com12 32 . . . 4 (𝑥𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹𝑥)))
594, 58vtoclga 3241 . . 3 (𝐶𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹𝐶)))
6059com12 32 . 2 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝐶𝐴𝐶 ⊆ (𝐹𝐶)))
61603impia 1252 1 ((𝐹 Fn 𝐴 ∧ Smo 𝐹𝐶𝐴) → 𝐶 ⊆ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2892  wss 3536  Ord word 5622  Oncon0 5623   Fn wfn 5782  cfv 5787  Smo wsmo 7303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-ord 5626  df-on 5627  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fv 5795  df-smo 7304
This theorem is referenced by:  smorndom  7326  oismo  8302
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