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Theorem ttukeylem5 9928
Description: Lemma for ttukey 9933. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
ttukeylem.2 (𝜑𝐵𝐴)
ttukeylem.3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
ttukeylem.4 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
Assertion
Ref Expression
ttukeylem5 ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))
Distinct variable groups:   𝑥,𝑧,𝐶   𝑥,𝐷   𝑥,𝐺,𝑧   𝜑,𝑧   𝑥,𝐴,𝑧   𝑥,𝐵,𝑧   𝑥,𝐹,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝐷(𝑧)

Proof of Theorem ttukeylem5
Dummy variables 𝑎 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq2 3944 . . . . . 6 (𝑦 = 𝑎 → (𝐶𝑦𝐶𝑎))
2 fveq2 6649 . . . . . . 7 (𝑦 = 𝑎 → (𝐺𝑦) = (𝐺𝑎))
32sseq2d 3950 . . . . . 6 (𝑦 = 𝑎 → ((𝐺𝐶) ⊆ (𝐺𝑦) ↔ (𝐺𝐶) ⊆ (𝐺𝑎)))
41, 3imbi12d 348 . . . . 5 (𝑦 = 𝑎 → ((𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)) ↔ (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))))
54imbi2d 344 . . . 4 (𝑦 = 𝑎 → (((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))) ↔ ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)))))
6 sseq2 3944 . . . . . 6 (𝑦 = 𝐷 → (𝐶𝑦𝐶𝐷))
7 fveq2 6649 . . . . . . 7 (𝑦 = 𝐷 → (𝐺𝑦) = (𝐺𝐷))
87sseq2d 3950 . . . . . 6 (𝑦 = 𝐷 → ((𝐺𝐶) ⊆ (𝐺𝑦) ↔ (𝐺𝐶) ⊆ (𝐺𝐷)))
96, 8imbi12d 348 . . . . 5 (𝑦 = 𝐷 → ((𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)) ↔ (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷))))
109imbi2d 344 . . . 4 (𝑦 = 𝐷 → (((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))) ↔ ((𝜑𝐶 ∈ On) → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷)))))
11 r19.21v 3145 . . . . 5 (∀𝑎𝑦 ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) ↔ ((𝜑𝐶 ∈ On) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))))
12 onsseleq 6204 . . . . . . . . . 10 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶𝑦 ↔ (𝐶𝑦𝐶 = 𝑦)))
1312ad4ant23 752 . . . . . . . . 9 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 ↔ (𝐶𝑦𝐶 = 𝑦)))
14 sseq2 3944 . . . . . . . . . . . . 13 (if(𝑦 = ∅, 𝐵, (𝐺𝑦)) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))) → ((𝐺𝐶) ⊆ if(𝑦 = ∅, 𝐵, (𝐺𝑦)) ↔ (𝐺𝐶) ⊆ if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)))))
15 sseq2 3944 . . . . . . . . . . . . 13 (((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))) → ((𝐺𝐶) ⊆ ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)) ↔ (𝐺𝐶) ⊆ if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)))))
16 ttukeylem.4 . . . . . . . . . . . . . . . . . 18 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
1716tfr1 8020 . . . . . . . . . . . . . . . . 17 𝐺 Fn On
18 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝑦 ∈ On)
19 onss 7489 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On → 𝑦 ⊆ On)
2018, 19syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝑦 ⊆ On)
21 simprr 772 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → 𝐶𝑦)
22 fnfvima 6977 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn On ∧ 𝑦 ⊆ On ∧ 𝐶𝑦) → (𝐺𝐶) ∈ (𝐺𝑦))
2317, 20, 21, 22mp3an2i 1463 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ∈ (𝐺𝑦))
24 elssuni 4833 . . . . . . . . . . . . . . . 16 ((𝐺𝐶) ∈ (𝐺𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦))
2523, 24syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ (𝐺𝑦))
26 n0i 4252 . . . . . . . . . . . . . . . 16 (𝐶𝑦 → ¬ 𝑦 = ∅)
27 iffalse 4437 . . . . . . . . . . . . . . . 16 𝑦 = ∅ → if(𝑦 = ∅, 𝐵, (𝐺𝑦)) = (𝐺𝑦))
2821, 26, 273syl 18 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → if(𝑦 = ∅, 𝐵, (𝐺𝑦)) = (𝐺𝑦))
2925, 28sseqtrrd 3959 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ if(𝑦 = ∅, 𝐵, (𝐺𝑦)))
3029adantr 484 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ if(𝑦 = ∅, 𝐵, (𝐺𝑦)))
3121adantr 484 . . . . . . . . . . . . . . . 16 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝐶𝑦)
32 elssuni 4833 . . . . . . . . . . . . . . . 16 (𝐶𝑦𝐶 𝑦)
3331, 32syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝐶 𝑦)
34 sseq2 3944 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝐶𝑎𝐶 𝑦))
35 fveq2 6649 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑦 → (𝐺𝑎) = (𝐺 𝑦))
3635sseq2d 3950 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → ((𝐺𝐶) ⊆ (𝐺𝑎) ↔ (𝐺𝐶) ⊆ (𝐺 𝑦)))
3734, 36imbi12d 348 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → ((𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ↔ (𝐶 𝑦 → (𝐺𝐶) ⊆ (𝐺 𝑦))))
38 simplrl 776 . . . . . . . . . . . . . . . 16 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)))
39 vuniex 7449 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
4039sucid 6242 . . . . . . . . . . . . . . . . 17 𝑦 ∈ suc 𝑦
41 eloni 6173 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ On → Ord 𝑦)
42 orduniorsuc 7529 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑦 → (𝑦 = 𝑦𝑦 = suc 𝑦))
4318, 41, 423syl 18 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝑦 = 𝑦𝑦 = suc 𝑦))
4443orcanai 1000 . . . . . . . . . . . . . . . . 17 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝑦 = suc 𝑦)
4540, 44eleqtrrid 2900 . . . . . . . . . . . . . . . 16 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → 𝑦𝑦)
4637, 38, 45rspcdva 3576 . . . . . . . . . . . . . . 15 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → (𝐶 𝑦 → (𝐺𝐶) ⊆ (𝐺 𝑦)))
4733, 46mpd 15 . . . . . . . . . . . . . 14 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ (𝐺 𝑦))
48 ssun1 4102 . . . . . . . . . . . . . 14 (𝐺 𝑦) ⊆ ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))
4947, 48sstrdi 3930 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) ∧ ¬ 𝑦 = 𝑦) → (𝐺𝐶) ⊆ ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅)))
5014, 15, 30, 49ifbothda 4465 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))))
51 ttukeylem.1 . . . . . . . . . . . . . 14 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
52 ttukeylem.2 . . . . . . . . . . . . . 14 (𝜑𝐵𝐴)
53 ttukeylem.3 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
5451, 52, 53, 16ttukeylem3 9926 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ On) → (𝐺𝑦) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))))
5554ad4ant13 750 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝑦) = if(𝑦 = 𝑦, if(𝑦 = ∅, 𝐵, (𝐺𝑦)), ((𝐺 𝑦) ∪ if(((𝐺 𝑦) ∪ {(𝐹 𝑦)}) ∈ 𝐴, {(𝐹 𝑦)}, ∅))))
5650, 55sseqtrrd 3959 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) ∧ 𝐶𝑦)) → (𝐺𝐶) ⊆ (𝐺𝑦))
5756expr 460 . . . . . . . . . 10 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
58 fveq2 6649 . . . . . . . . . . . 12 (𝐶 = 𝑦 → (𝐺𝐶) = (𝐺𝑦))
59 eqimss 3974 . . . . . . . . . . . 12 ((𝐺𝐶) = (𝐺𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦))
6058, 59syl 17 . . . . . . . . . . 11 (𝐶 = 𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))
6160a1i 11 . . . . . . . . . 10 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶 = 𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
6257, 61jaod 856 . . . . . . . . 9 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝐶𝑦𝐶 = 𝑦) → (𝐺𝐶) ⊆ (𝐺𝑦)))
6313, 62sylbid 243 . . . . . . . 8 ((((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) ∧ ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))
6463ex 416 . . . . . . 7 (((𝜑𝐶 ∈ On) ∧ 𝑦 ∈ On) → (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦))))
6564expcom 417 . . . . . 6 (𝑦 ∈ On → ((𝜑𝐶 ∈ On) → (∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎)) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
6665a2d 29 . . . . 5 (𝑦 ∈ On → (((𝜑𝐶 ∈ On) → ∀𝑎𝑦 (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
6711, 66syl5bi 245 . . . 4 (𝑦 ∈ On → (∀𝑎𝑦 ((𝜑𝐶 ∈ On) → (𝐶𝑎 → (𝐺𝐶) ⊆ (𝐺𝑎))) → ((𝜑𝐶 ∈ On) → (𝐶𝑦 → (𝐺𝐶) ⊆ (𝐺𝑦)))))
685, 10, 67tfis3 7556 . . 3 (𝐷 ∈ On → ((𝜑𝐶 ∈ On) → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷))))
6968expdcom 418 . 2 (𝜑 → (𝐶 ∈ On → (𝐷 ∈ On → (𝐶𝐷 → (𝐺𝐶) ⊆ (𝐺𝐷)))))
70693imp2 1346 1 ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084  wal 1536   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  ifcif 4428  𝒫 cpw 4500  {csn 4528   cuni 4803  cmpt 5113  dom cdm 5523  ran crn 5524  cima 5526  Ord word 6162  Oncon0 6163  suc csuc 6165   Fn wfn 6323  1-1-ontowf1o 6327  cfv 6328  recscrecs 7994  Fincfn 8496  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-wrecs 7934  df-recs 7995
This theorem is referenced by:  ttukeylem6  9929  ttukeylem7  9930
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