Theorem ranksnb 9867

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.

Step	Hyp	Ref	Expression
1		fveq2 6906	. . . . . 6 ⊢ (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
2	1	eleq1d 2826	. . . . 5 ⊢ (𝑦 = 𝐴 → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
3	2	ralsng 4675	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
4	3	rabbidv 3444	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
5	4	inteqd 4951	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
6		snwf 9849	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
7		rankval3b 9866	. . 3 ⊢ ({𝐴} ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥})
8	6, 7	syl 17	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥})
9		rankon 9835	. . 3 ⊢ (rank‘𝐴) ∈ On
10		onsucmin 7841	. . 3 ⊢ ((rank‘𝐴) ∈ On → suc (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
11	9, 10	mp1i 13	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
12	5, 8, 11	3eqtr4d 2787	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436  {csn 4626  ∪ cuni 4907  ∩ cint 4946   “ cima 5688  Oncon0 6384  suc csuc 6386  ‘cfv 6561  𝑅₁cr1 9802  rankcrnk 9803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805

This theorem is referenced by:  rankprb  9891  ranksn  9894  rankcf  10817  rankaltopb  35980