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Theorem ranksnb 8854
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 6330 . . . . . 6 (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
21eleq1d 2835 . . . . 5 (𝑦 = 𝐴 → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
32ralsng 4356 . . . 4 (𝐴 (𝑅1 “ On) → (∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
43rabbidv 3339 . . 3 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
54inteqd 4616 . 2 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
6 snwf 8836 . . 3 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
7 rankval3b 8853 . . 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 8822 . . 3 (rank‘𝐴) ∈ On
10 onsucmin 7168 . . 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 2815 1 (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wral 3061  {crab 3065  {csn 4316   cuni 4574   cint 4611  cima 5252  Oncon0 5864  suc csuc 5866  cfv 6029  𝑅1cr1 8789  rankcrnk 8790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-r1 8791  df-rank 8792
This theorem is referenced by:  rankprb  8878  ranksn  8881  rankcf  9801  rankaltopb  32419
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