Theorem rankval2b 35359

Description: Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. This variant of rankval2 9773 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅₁. (Contributed by BTernaryTau, 19-Jan-2026.)

Assertion

Ref	Expression
rankval2b	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)})

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankval2b

Step	Hyp	Ref	Expression
1		rankvalb 9752	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
2		r1suc 9725	. . . . . 6 ⊢ (𝑥 ∈ On → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
3	2	eleq2d 2847	. . . . 5 ⊢ (𝑥 ∈ On → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ 𝒫 (𝑅1‘𝑥)))
4		fvex 6876	. . . . . 6 ⊢ (𝑅1‘𝑥) ∈ V
5	4	elpw2 5289	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝐴 ⊆ (𝑅1‘𝑥))
6	3, 5	bitrdi 289	. . . 4 ⊢ (𝑥 ∈ On → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ⊆ (𝑅1‘𝑥)))
7	6	rabbiia 3417	. . 3 ⊢ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)}
8	7	inteqi 4908	. 2 ⊢ ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)}
9	1, 8	eqtrdi 2812	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)})

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  {crab 3413   ⊆ wss 3904  𝒫 cpw 4554  ∪ cuni 4864  ∩ cint 4904   “ cima 5648  Oncon0 6342  suc csuc 6344  ‘cfv 6517  𝑅₁cr1 9717  rankcrnk 9718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-r1 9719  df-rank 9720

This theorem is referenced by:  rankval4b  35360