Theorem r1tr 9783

Description: Each stage of the cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
r1tr	⊢ Tr (𝑅1‘𝐴)

Proof of Theorem r1tr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1dmlim 9772	. . . . 5 ⊢ Lim dom 𝑅1
2		limord 6424	. . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
3		ordsson 7797	. . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
4	1, 2, 3	mp2b 10	. . . 4 ⊢ dom 𝑅1 ⊆ On
5	4	sseli 3904	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
6		fveq2 6885	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
7		r10 9775	. . . . . 6 ⊢ (𝑅1‘∅) = ∅
8	6, 7	eqtrdi 2811	. . . . 5 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = ∅)
9		treq 5218	. . . . 5 ⊢ ((𝑅1‘𝑥) = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
10	8, 9	syl 18	. . . 4 ⊢ (𝑥 = ∅ → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
11		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
12		treq 5218	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝑦)))
13	11, 12	syl 18	. . . 4 ⊢ (𝑥 = 𝑦 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝑦)))
14		fveq2 6885	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
15		treq 5218	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘suc 𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘suc 𝑦)))
16	14, 15	syl 18	. . . 4 ⊢ (𝑥 = suc 𝑦 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘suc 𝑦)))
17		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅1‘𝑥) = (𝑅1‘𝐴))
18		treq 5218	. . . . 5 ⊢ ((𝑅1‘𝑥) = (𝑅1‘𝐴) → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝐴)))
19	17, 18	syl 18	. . . 4 ⊢ (𝑥 = 𝐴 → (Tr (𝑅1‘𝑥) ↔ Tr (𝑅1‘𝐴)))
20		tr0 5224	. . . 4 ⊢ Tr ∅
21		limsuc 7860	. . . . . . . 8 ⊢ (Lim dom 𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
22	1, 21	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1)
23		pwtr 5419	. . . . . . . . 9 ⊢ (Tr (𝑅1‘𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦))
24	23	bilani 510	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr 𝒫 (𝑅1‘𝑦))
25		r1sucg 9776	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦))
26		treq 5218	. . . . . . . . 9 ⊢ ((𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦) → (Tr (𝑅1‘suc 𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦)))
27	25, 26	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr 𝒫 (𝑅1‘𝑦)))
28	24, 27	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → (𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
29	22, 28	biimtrrid 246	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → (suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦)))
30		ndmfv 6917	. . . . . . . 8 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → (𝑅1‘suc 𝑦) = ∅)
31		treq 5218	. . . . . . . 8 ⊢ ((𝑅1‘suc 𝑦) = ∅ → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
32	30, 31	syl 18	. . . . . . 7 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → (Tr (𝑅1‘suc 𝑦) ↔ Tr ∅))
33	20, 32	mpbiri 261	. . . . . 6 ⊢ (¬ suc 𝑦 ∈ dom 𝑅1 → Tr (𝑅1‘suc 𝑦))
34	29, 33	pm2.61d1 182	. . . . 5 ⊢ ((𝑦 ∈ On ∧ Tr (𝑅1‘𝑦)) → Tr (𝑅1‘suc 𝑦))
35	34	ex 418	. . . 4 ⊢ (𝑦 ∈ On → (Tr (𝑅1‘𝑦) → Tr (𝑅1‘suc 𝑦)))
36		triun 5226	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
37		r1limg 9778	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
38	37	ancoms 464	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦))
39		treq 5218	. . . . . . . . 9 ⊢ ((𝑅1‘𝑥) = ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦) → (Tr (𝑅1‘𝑥) ↔ Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)))
40	38, 39	syl 18	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (Tr (𝑅1‘𝑥) ↔ Tr ∪ 𝑦 ∈ 𝑥 (𝑅1‘𝑦)))
41	36, 40	imbitrrid 249	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr (𝑅1‘𝑥)))
42	41	impancom 457	. . . . . 6 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦)) → (𝑥 ∈ dom 𝑅1 → Tr (𝑅1‘𝑥)))
43		ndmfv 6917	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → (𝑅1‘𝑥) = ∅)
44	43, 9	syl 18	. . . . . . 7 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → (Tr (𝑅1‘𝑥) ↔ Tr ∅))
45	20, 44	mpbiri 261	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom 𝑅1 → Tr (𝑅1‘𝑥))
46	42, 45	pm2.61d1 182	. . . . 5 ⊢ ((Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦)) → Tr (𝑅1‘𝑥))
47	46	ex 418	. . . 4 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 Tr (𝑅1‘𝑦) → Tr (𝑅1‘𝑥)))
48	10, 13, 16, 19, 20, 35, 47	tfinds 7871	. . 3 ⊢ (𝐴 ∈ On → Tr (𝑅1‘𝐴))
49	5, 48	syl 18	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
50		ndmfv 6917	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∅)
51		treq 5218	. . . 4 ⊢ ((𝑅1‘𝐴) = ∅ → (Tr (𝑅1‘𝐴) ↔ Tr ∅))
52	50, 51	syl 18	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → (Tr (𝑅1‘𝐴) ↔ Tr ∅))
53	20, 52	mpbiri 261	. 2 ⊢ (¬ 𝐴 ∈ dom 𝑅1 → Tr (𝑅1‘𝐴))
54	49, 53	pm2.61i 184	1 ⊢ Tr (𝑅1‘𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556  ∪ ciun 4950  Tr wtr 5211  dom cdm 5650  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364  ‘cfv 6538  𝑅₁cr1 9766

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768

This theorem is used by:  r1tr2  9784  r1ordg  9785  r1ord3g  9786  r1ord2  9788  r1sssuc  9790  r1pwss  9791  r1val1  9793  rankwflemb  9800  rankwflembOLD  9801  r1elwf  9804  r1elssi  9813  uniwf  9828  tcrank  9901  elhf4  9912  ackbij2lem3  10318  r1limwun  10821  tskhf  10853  r1filim  35729  ttcwf  37312  inagrud  45280