Theorem onsetrec 42250

Description: Construct On using set recursion. When 𝑥 ∈ On, the function 𝐹 constructs the least ordinal greater than any of the elements of 𝑥, which is ∪ 𝑥 for a limit ordinal and suc ∪ 𝑥 for a successor ordinal.

For example, (𝐹‘{1_𝑜, 2_𝑜}) = {∪ {1_𝑜, 2_𝑜}, suc ∪ {1_𝑜, 2_𝑜}} = {2_𝑜, 3_𝑜} which contains 3_𝑜, and (𝐹‘ω) = {∪ ω, suc ∪ ω} = {ω, ω +_𝑜 1_𝑜}, which contains ω. If we start with the empty set and keep applying 𝐹 transfinitely many times, all ordinal numbers will be generated.

Any function 𝐹 fulfilling lemmas onsetreclem2 42248 and onsetreclem3 42249 will recursively generate On; for example, 𝐹 = (𝑥 ∈ V ↦ suc suc ∪ 𝑥}) also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of On is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝑥 ∣ Tr 𝑦}), based on dfon2 30941.

The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable condition between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 22-Jun-2021.)

Hypothesis

Ref	Expression
onsetrec.1	⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})

Assertion

Ref	Expression
onsetrec	⊢ setrecs(𝐹) = On

Proof of Theorem onsetrec

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . 4 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		onsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
3	2	onsetreclem2 42248	. . . . . 6 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)
4	3	ax-gen 1713	. . . . 5 ⊢ ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)
5	4	a1i 11	. . . 4 ⊢ (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On))
6	1, 5	setrec2v 42242	. . 3 ⊢ (⊤ → setrecs(𝐹) ⊆ On)
7	6	trud 1484	. 2 ⊢ setrecs(𝐹) ⊆ On
8		vex 3176	. . . . . . 7 ⊢ 𝑎 ∈ V
9	8	a1i 11	. . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
10		id 22	. . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
11	1, 9, 10	setrec1 42237	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
12	2	onsetreclem3 42249	. . . . 5 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))
13		ssel 3562	. . . . 5 ⊢ ((𝐹‘𝑎) ⊆ setrecs(𝐹) → (𝑎 ∈ (𝐹‘𝑎) → 𝑎 ∈ setrecs(𝐹)))
14	11, 12, 13	syl2im 39	. . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ On → 𝑎 ∈ setrecs(𝐹)))
15	14	com12 32	. . 3 ⊢ (𝑎 ∈ On → (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)))
16	15	rgen 2906	. 2 ⊢ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
17		tfi 6945	. 2 ⊢ ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On)
18	7, 16, 17	mp2an 704	1 ⊢ setrecs(𝐹) = On

Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  {cpr 4127  ∪ cuni 4372   ↦ cmpt 4643  Oncon0 5640  suc csuc 5642  ‘cfv 5804  setrecscsetrecs 42229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511  df-setrecs 42230

This theorem is referenced by: (None)