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Theorem ranksnb 8573
 Description: The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
ranksnb (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))

Proof of Theorem ranksnb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . . . 6 (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
21eleq1d 2672 . . . . 5 (𝑦 = 𝐴 → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
32ralsng 4165 . . . 4 (𝐴 (𝑅1 “ On) → (∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
43rabbidv 3164 . . 3 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
54inteqd 4415 . 2 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
6 snwf 8555 . . 3 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
7 rankval3b 8572 . . 3 ({𝐴} ∈ (𝑅1 “ On) → (rank‘{𝐴}) = {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥})
86, 7syl 17 . 2 (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥})
9 rankon 8541 . . 3 (rank‘𝐴) ∈ On
10 onsucmin 6913 . . 3 ((rank‘𝐴) ∈ On → suc (rank‘𝐴) = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
119, 10mp1i 13 . 2 (𝐴 (𝑅1 “ On) → suc (rank‘𝐴) = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
125, 8, 113eqtr4d 2654 1 (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  {csn 4125  ∪ cuni 4372  ∩ cint 4410   “ cima 5041  Oncon0 5640  suc csuc 5642  ‘cfv 5804  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511 This theorem is referenced by:  rankprb  8597  ranksn  8600  rankcf  9478  rankaltopb  31256
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