Theorem ranksnb 9742

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 6827	. . . . . 6 ⊢ (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
2	1	eleq1d 2824	. . . . 5 ⊢ (𝑦 = 𝐴 → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
3	2	ralsng 4607	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
4	3	rabbidv 3398	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
5	4	inteqd 4882	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
6		snwf 9724	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
7		rankval3b 9741	. . 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 9710	. . 3 ⊢ (rank‘𝐴) ∈ On
10		onsucmin 7761	. . 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 2784	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391  {csn 4555  ∪ cuni 4838  ∩ cint 4877   “ cima 5621  Oncon0 6310  suc csuc 6312  ‘cfv 6485  𝑅₁cr1 9677  rankcrnk 9678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9679  df-rank 9680

This theorem is referenced by:  rankprb  9766  ranksn  9769  rankcf  10691  rankaltopb  36207