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Theorem tfisi 7221
Description: A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Hypotheses
Ref Expression
tfisi.a (𝜑𝐴𝑉)
tfisi.b (𝜑𝑇 ∈ On)
tfisi.c ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅𝑇) ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
tfisi.d (𝑥 = 𝑦 → (𝜓𝜒))
tfisi.e (𝑥 = 𝐴 → (𝜓𝜃))
tfisi.f (𝑥 = 𝑦𝑅 = 𝑆)
tfisi.g (𝑥 = 𝐴𝑅 = 𝑇)
Assertion
Ref Expression
tfisi (𝜑𝜃)
Distinct variable groups:   𝑥,𝑦,𝑇   𝑦,𝑅   𝑥,𝑆   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦   𝑥,𝐴   𝜃,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem tfisi
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3763 . 2 𝑇𝑇
2 eqid 2758 . . . . 5 𝑇 = 𝑇
3 tfisi.a . . . . . 6 (𝜑𝐴𝑉)
4 tfisi.b . . . . . . 7 (𝜑𝑇 ∈ On)
5 eqeq2 2769 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑅 = 𝑧𝑅 = 𝑤))
6 sseq1 3765 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → (𝑧𝑇𝑤𝑇))
76anbi2d 742 . . . . . . . . . . . 12 (𝑧 = 𝑤 → ((𝜑𝑧𝑇) ↔ (𝜑𝑤𝑇)))
87imbi1d 330 . . . . . . . . . . 11 (𝑧 = 𝑤 → (((𝜑𝑧𝑇) → 𝜓) ↔ ((𝜑𝑤𝑇) → 𝜓)))
95, 8imbi12d 333 . . . . . . . . . 10 (𝑧 = 𝑤 → ((𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ (𝑅 = 𝑤 → ((𝜑𝑤𝑇) → 𝜓))))
109albidv 1996 . . . . . . . . 9 (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑤 → ((𝜑𝑤𝑇) → 𝜓))))
11 tfisi.f . . . . . . . . . . . 12 (𝑥 = 𝑦𝑅 = 𝑆)
1211eqeq1d 2760 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑅 = 𝑤𝑆 = 𝑤))
13 tfisi.d . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝜓𝜒))
1413imbi2d 329 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝜑𝑤𝑇) → 𝜓) ↔ ((𝜑𝑤𝑇) → 𝜒)))
1512, 14imbi12d 333 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑅 = 𝑤 → ((𝜑𝑤𝑇) → 𝜓)) ↔ (𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))))
1615cbvalv 2416 . . . . . . . . 9 (∀𝑥(𝑅 = 𝑤 → ((𝜑𝑤𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)))
1710, 16syl6bb 276 . . . . . . . 8 (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))))
18 eqeq2 2769 . . . . . . . . . 10 (𝑧 = 𝑇 → (𝑅 = 𝑧𝑅 = 𝑇))
19 sseq1 3765 . . . . . . . . . . . 12 (𝑧 = 𝑇 → (𝑧𝑇𝑇𝑇))
2019anbi2d 742 . . . . . . . . . . 11 (𝑧 = 𝑇 → ((𝜑𝑧𝑇) ↔ (𝜑𝑇𝑇)))
2120imbi1d 330 . . . . . . . . . 10 (𝑧 = 𝑇 → (((𝜑𝑧𝑇) → 𝜓) ↔ ((𝜑𝑇𝑇) → 𝜓)))
2218, 21imbi12d 333 . . . . . . . . 9 (𝑧 = 𝑇 → ((𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ (𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓))))
2322albidv 1996 . . . . . . . 8 (𝑧 = 𝑇 → (∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓))))
24 simp3l 1244 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝜑)
25 simp2 1132 . . . . . . . . . . . . 13 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑅 = 𝑧)
26 simp1l 1240 . . . . . . . . . . . . 13 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑧 ∈ On)
2725, 26eqeltrd 2837 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑅 ∈ On)
28 simp3r 1245 . . . . . . . . . . . . 13 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑧𝑇)
2925, 28eqsstrd 3778 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑅𝑇)
30 simpl3l 1287 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝜑)
31 simpl1l 1279 . . . . . . . . . . . . . . . . . 18 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑧 ∈ On)
32 simpr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅𝑅)
33 simpl2 1230 . . . . . . . . . . . . . . . . . . 19 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑅 = 𝑧)
3432, 33eleqtrd 2839 . . . . . . . . . . . . . . . . . 18 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅𝑧)
35 onelss 5925 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ On → (𝑣 / 𝑥𝑅𝑧𝑣 / 𝑥𝑅𝑧))
3631, 34, 35sylc 65 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅𝑧)
37 simpl3r 1289 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑧𝑇)
3836, 37sstrd 3752 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅𝑇)
39 simpl1r 1281 . . . . . . . . . . . . . . . . . 18 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)))
40 eqeq2 2769 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑣 / 𝑥𝑅 → (𝑆 = 𝑤𝑆 = 𝑣 / 𝑥𝑅))
41 sseq1 3765 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑣 / 𝑥𝑅 → (𝑤𝑇𝑣 / 𝑥𝑅𝑇))
4241anbi2d 742 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑣 / 𝑥𝑅 → ((𝜑𝑤𝑇) ↔ (𝜑𝑣 / 𝑥𝑅𝑇)))
4342imbi1d 330 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑣 / 𝑥𝑅 → (((𝜑𝑤𝑇) → 𝜒) ↔ ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)))
4440, 43imbi12d 333 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑣 / 𝑥𝑅 → ((𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)) ↔ (𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒))))
4544albidv 1996 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑣 / 𝑥𝑅 → (∀𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)) ↔ ∀𝑦(𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒))))
4645rspcva 3445 . . . . . . . . . . . . . . . . . 18 ((𝑣 / 𝑥𝑅𝑧 ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) → ∀𝑦(𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)))
4734, 39, 46syl2anc 696 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → ∀𝑦(𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)))
48 eqidd 2759 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅 = 𝑣 / 𝑥𝑅)
49 nfcv 2900 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥𝑦
50 nfcv 2900 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥𝑆
5149, 50, 11csbhypf 3691 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑦𝑣 / 𝑥𝑅 = 𝑆)
5251eqcomd 2764 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑦𝑆 = 𝑣 / 𝑥𝑅)
5352equcoms 2100 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑣𝑆 = 𝑣 / 𝑥𝑅)
5453eqeq1d 2760 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (𝑆 = 𝑣 / 𝑥𝑅𝑣 / 𝑥𝑅 = 𝑣 / 𝑥𝑅))
55 nfv 1990 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥𝜒
5655, 13sbhypf 3391 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑦 → ([𝑣 / 𝑥]𝜓𝜒))
5756bicomd 213 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑦 → (𝜒 ↔ [𝑣 / 𝑥]𝜓))
5857equcoms 2100 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑣 → (𝜒 ↔ [𝑣 / 𝑥]𝜓))
5958imbi2d 329 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒) ↔ ((𝜑𝑣 / 𝑥𝑅𝑇) → [𝑣 / 𝑥]𝜓)))
6054, 59imbi12d 333 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑣 → ((𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)) ↔ (𝑣 / 𝑥𝑅 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → [𝑣 / 𝑥]𝜓))))
6160spv 2403 . . . . . . . . . . . . . . . . 17 (∀𝑦(𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)) → (𝑣 / 𝑥𝑅 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → [𝑣 / 𝑥]𝜓)))
6247, 48, 61sylc 65 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → ((𝜑𝑣 / 𝑥𝑅𝑇) → [𝑣 / 𝑥]𝜓))
6330, 38, 62mp2and 717 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → [𝑣 / 𝑥]𝜓)
6463ex 449 . . . . . . . . . . . . . 14 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → (𝑣 / 𝑥𝑅𝑅 → [𝑣 / 𝑥]𝜓))
6564alrimiv 2002 . . . . . . . . . . . . 13 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → ∀𝑣(𝑣 / 𝑥𝑅𝑅 → [𝑣 / 𝑥]𝜓))
6651eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → (𝑣 / 𝑥𝑅𝑅𝑆𝑅))
6766, 56imbi12d 333 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ((𝑣 / 𝑥𝑅𝑅 → [𝑣 / 𝑥]𝜓) ↔ (𝑆𝑅𝜒)))
6867cbvalv 2416 . . . . . . . . . . . . 13 (∀𝑣(𝑣 / 𝑥𝑅𝑅 → [𝑣 / 𝑥]𝜓) ↔ ∀𝑦(𝑆𝑅𝜒))
6965, 68sylib 208 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → ∀𝑦(𝑆𝑅𝜒))
70 tfisi.c . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅𝑇) ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
7124, 27, 29, 69, 70syl121anc 1482 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝜓)
72713exp 1113 . . . . . . . . . 10 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) → (𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)))
7372alrimiv 2002 . . . . . . . . 9 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) → ∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)))
7473ex 449 . . . . . . . 8 (𝑧 ∈ On → (∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)) → ∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓))))
7517, 23, 74tfis3 7220 . . . . . . 7 (𝑇 ∈ On → ∀𝑥(𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓)))
764, 75syl 17 . . . . . 6 (𝜑 → ∀𝑥(𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓)))
77 tfisi.g . . . . . . . . 9 (𝑥 = 𝐴𝑅 = 𝑇)
7877eqeq1d 2760 . . . . . . . 8 (𝑥 = 𝐴 → (𝑅 = 𝑇𝑇 = 𝑇))
79 tfisi.e . . . . . . . . 9 (𝑥 = 𝐴 → (𝜓𝜃))
8079imbi2d 329 . . . . . . . 8 (𝑥 = 𝐴 → (((𝜑𝑇𝑇) → 𝜓) ↔ ((𝜑𝑇𝑇) → 𝜃)))
8178, 80imbi12d 333 . . . . . . 7 (𝑥 = 𝐴 → ((𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓)) ↔ (𝑇 = 𝑇 → ((𝜑𝑇𝑇) → 𝜃))))
8281spcgv 3431 . . . . . 6 (𝐴𝑉 → (∀𝑥(𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓)) → (𝑇 = 𝑇 → ((𝜑𝑇𝑇) → 𝜃))))
833, 76, 82sylc 65 . . . . 5 (𝜑 → (𝑇 = 𝑇 → ((𝜑𝑇𝑇) → 𝜃)))
842, 83mpi 20 . . . 4 (𝜑 → ((𝜑𝑇𝑇) → 𝜃))
8584expd 451 . . 3 (𝜑 → (𝜑 → (𝑇𝑇𝜃)))
8685pm2.43i 52 . 2 (𝜑 → (𝑇𝑇𝜃))
871, 86mpi 20 1 (𝜑𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1628   = wceq 1630  [wsb 2044  wcel 2137  wral 3048  csb 3672  wss 3713  Oncon0 5882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-tr 4903  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-ord 5885  df-on 5886
This theorem is referenced by:  indcardi  9052
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