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Theorem rankval4 8682
Description: The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.)
Hypothesis
Ref Expression
rankr1b.1 𝐴 ∈ V
Assertion
Ref Expression
rankval4 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankval4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2761 . . . . . 6 𝑥𝐴
2 nfcv 2761 . . . . . . 7 𝑥𝑅1
3 nfiu1 4521 . . . . . . 7 𝑥 𝑥𝐴 suc (rank‘𝑥)
42, 3nffv 6160 . . . . . 6 𝑥(𝑅1 𝑥𝐴 suc (rank‘𝑥))
51, 4dfss2f 3578 . . . . 5 (𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝑅1 𝑥𝐴 suc (rank‘𝑥))))
6 vex 3192 . . . . . . 7 𝑥 ∈ V
76rankid 8648 . . . . . 6 𝑥 ∈ (𝑅1‘suc (rank‘𝑥))
8 ssiun2 4534 . . . . . . . 8 (𝑥𝐴 → suc (rank‘𝑥) ⊆ 𝑥𝐴 suc (rank‘𝑥))
9 rankon 8610 . . . . . . . . . 10 (rank‘𝑥) ∈ On
109onsuci 6992 . . . . . . . . 9 suc (rank‘𝑥) ∈ On
11 rankr1b.1 . . . . . . . . . 10 𝐴 ∈ V
1210rgenw 2919 . . . . . . . . . 10 𝑥𝐴 suc (rank‘𝑥) ∈ On
13 iunon 7388 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ∀𝑥𝐴 suc (rank‘𝑥) ∈ On) → 𝑥𝐴 suc (rank‘𝑥) ∈ On)
1411, 12, 13mp2an 707 . . . . . . . . 9 𝑥𝐴 suc (rank‘𝑥) ∈ On
15 r1ord3 8597 . . . . . . . . 9 ((suc (rank‘𝑥) ∈ On ∧ 𝑥𝐴 suc (rank‘𝑥) ∈ On) → (suc (rank‘𝑥) ⊆ 𝑥𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥))))
1610, 14, 15mp2an 707 . . . . . . . 8 (suc (rank‘𝑥) ⊆ 𝑥𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)))
178, 16syl 17 . . . . . . 7 (𝑥𝐴 → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)))
1817sseld 3586 . . . . . 6 (𝑥𝐴 → (𝑥 ∈ (𝑅1‘suc (rank‘𝑥)) → 𝑥 ∈ (𝑅1 𝑥𝐴 suc (rank‘𝑥))))
197, 18mpi 20 . . . . 5 (𝑥𝐴𝑥 ∈ (𝑅1 𝑥𝐴 suc (rank‘𝑥)))
205, 19mpgbir 1723 . . . 4 𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥))
21 fvex 6163 . . . . 5 (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ∈ V
2221rankss 8664 . . . 4 (𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) → (rank‘𝐴) ⊆ (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))))
2320, 22ax-mp 5 . . 3 (rank‘𝐴) ⊆ (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥)))
24 r1ord3 8597 . . . . . . 7 (( 𝑥𝐴 suc (rank‘𝑥) ∈ On ∧ 𝑦 ∈ On) → ( 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)))
2514, 24mpan 705 . . . . . 6 (𝑦 ∈ On → ( 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)))
2625ss2rabi 3668 . . . . 5 {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)}
27 intss 4468 . . . . 5 ({𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)} → {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)} ⊆ {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦})
2826, 27ax-mp 5 . . . 4 {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)} ⊆ {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦}
29 rankval2 8633 . . . . 5 ((𝑅1 𝑥𝐴 suc (rank‘𝑥)) ∈ V → (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))) = {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)})
3021, 29ax-mp 5 . . . 4 (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))) = {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)}
31 intmin 4467 . . . . . 6 ( 𝑥𝐴 suc (rank‘𝑥) ∈ On → {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} = 𝑥𝐴 suc (rank‘𝑥))
3214, 31ax-mp 5 . . . . 5 {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} = 𝑥𝐴 suc (rank‘𝑥)
3332eqcomi 2630 . . . 4 𝑥𝐴 suc (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦}
3428, 30, 333sstr4i 3628 . . 3 (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))) ⊆ 𝑥𝐴 suc (rank‘𝑥)
3523, 34sstri 3596 . 2 (rank‘𝐴) ⊆ 𝑥𝐴 suc (rank‘𝑥)
36 iunss 4532 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
3711rankel 8654 . . . 4 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
38 rankon 8610 . . . . 5 (rank‘𝐴) ∈ On
399, 38onsucssi 6995 . . . 4 ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘𝐴))
4037, 39sylib 208 . . 3 (𝑥𝐴 → suc (rank‘𝑥) ⊆ (rank‘𝐴))
4136, 40mprgbir 2922 . 2 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴)
4235, 41eqssi 3603 1 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  wss 3559   cint 4445   ciun 4490  Oncon0 5687  suc csuc 5689  cfv 5852  𝑅1cr1 8577  rankcrnk 8578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8449  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8579  df-rank 8580
This theorem is referenced by:  rankbnd  8683  rankc1  8685
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