Theorem ranksnb 9844

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 6890	. . . . . 6 ⊢ (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
2	1	eleq1d 2845	. . . . 5 ⊢ (𝑦 = 𝐴 → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
3	2	ralsng 4632	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
4	3	rabbidv 3419	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
5	4	inteqd 4911	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
6		snwf 9824	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
7		rankval3b 9843	. . 3 ⊢ ({𝐴} ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥})
8	6, 7	syl 18	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥})
9		rankon 9810	. . 3 ⊢ (rank‘𝐴) ∈ On
10		onsucmin 7836	. . 3 ⊢ ((rank‘𝐴) ∈ On → suc (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
11	9, 10	mp1i 14	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
12	5, 8, 11	3eqtr4d 2805	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  {csn 4584  ∪ cuni 4866  ∩ cint 4906   “ cima 5654  Oncon0 6365  suc csuc 6367  ‘cfv 6543  𝑅₁cr1 9773  rankcrnk 9774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-om 7882  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-r1 9775  df-rank 9776

This theorem is used by:  rankprb  9872  ranksn  9875  rankcf  10869  rankaltopb  36840