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Theorem rankval4 9280
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 2980 . . . . . 6 𝑥𝐴
2 nfcv 2980 . . . . . . 7 𝑥𝑅1
3 nfiu1 4934 . . . . . . 7 𝑥 𝑥𝐴 suc (rank‘𝑥)
42, 3nffv 6661 . . . . . 6 𝑥(𝑅1 𝑥𝐴 suc (rank‘𝑥))
51, 4dfss2f 3942 . . . . 5 (𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝑅1 𝑥𝐴 suc (rank‘𝑥))))
6 vex 3482 . . . . . . 7 𝑥 ∈ V
76rankid 9246 . . . . . 6 𝑥 ∈ (𝑅1‘suc (rank‘𝑥))
8 ssiun2 4952 . . . . . . . 8 (𝑥𝐴 → suc (rank‘𝑥) ⊆ 𝑥𝐴 suc (rank‘𝑥))
9 rankon 9208 . . . . . . . . . 10 (rank‘𝑥) ∈ On
109onsuci 7536 . . . . . . . . 9 suc (rank‘𝑥) ∈ On
11 rankr1b.1 . . . . . . . . . 10 𝐴 ∈ V
1210rgenw 3144 . . . . . . . . . 10 𝑥𝐴 suc (rank‘𝑥) ∈ On
13 iunon 7959 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ∀𝑥𝐴 suc (rank‘𝑥) ∈ On) → 𝑥𝐴 suc (rank‘𝑥) ∈ On)
1411, 12, 13mp2an 691 . . . . . . . . 9 𝑥𝐴 suc (rank‘𝑥) ∈ On
15 r1ord3 9195 . . . . . . . . 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 691 . . . . . . . 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 3950 . . . . . 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 1801 . . . 4 𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥))
21 fvex 6664 . . . . 5 (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ∈ V
2221rankss 9262 . . . 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 9195 . . . . . . 7 (( 𝑥𝐴 suc (rank‘𝑥) ∈ On ∧ 𝑦 ∈ On) → ( 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)))
2514, 24mpan 689 . . . . . 6 (𝑦 ∈ On → ( 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)))
2625ss2rabi 4037 . . . . 5 {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)}
27 intss 4878 . . . . 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 9231 . . . . 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 4877 . . . . . 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 2833 . . . 4 𝑥𝐴 suc (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦}
3428, 30, 333sstr4i 3994 . . 3 (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))) ⊆ 𝑥𝐴 suc (rank‘𝑥)
3523, 34sstri 3960 . 2 (rank‘𝐴) ⊆ 𝑥𝐴 suc (rank‘𝑥)
36 iunss 4950 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
3711rankel 9252 . . . 4 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
38 rankon 9208 . . . . 5 (rank‘𝐴) ∈ On
399, 38onsucssi 7539 . . . 4 ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘𝐴))
4037, 39sylib 221 . . 3 (𝑥𝐴 → suc (rank‘𝑥) ⊆ (rank‘𝐴))
4136, 40mprgbir 3147 . 2 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴)
4235, 41eqssi 3967 1 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wral 3132  {crab 3136  Vcvv 3479  wss 3918   cint 4857   ciun 4900  Oncon0 6172  suc csuc 6174  cfv 6336  𝑅1cr1 9175  rankcrnk 9176
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-reg 9040  ax-inf2 9088
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7564  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-r1 9177  df-rank 9178
This theorem is referenced by:  rankbnd  9281  rankc1  9283  scottrankd  40792
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