Theorem ranksnb 9824

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 6885	. . . . . 6 ⊢ (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
2	1	eleq1d 2812	. . . . 5 ⊢ (𝑦 = 𝐴 → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
3	2	ralsng 4672	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
4	3	rabbidv 3434	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
5	4	inteqd 4948	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
6		snwf 9806	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
7		rankval3b 9823	. . 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 9792	. . 3 ⊢ (rank‘𝐴) ∈ On
10		onsucmin 7806	. . 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 2776	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3055  {crab 3426  {csn 4623  ∪ cuni 4902  ∩ cint 4943   “ cima 5672  Oncon0 6358  suc csuc 6360  ‘cfv 6537  𝑅₁cr1 9759  rankcrnk 9760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-r1 9761  df-rank 9762

This theorem is referenced by:  rankprb  9848  ranksn  9851  rankcf  10774  rankaltopb  35484