Theorem onsetrec 49690

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 49688 and onsetreclem3 49689 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 35773.

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 2729	. . . 4 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		onsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
3	2	onsetreclem2 49688	. . . . . 6 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)
4	3	ax-gen 1795	. . . . 5 ⊢ ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)
5	4	a1i 11	. . . 4 ⊢ (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On))
6	1, 5	setrec2v 49678	. . 3 ⊢ (⊤ → setrecs(𝐹) ⊆ On)
7	6	mptru 1547	. 2 ⊢ setrecs(𝐹) ⊆ On
8		vex 3448	. . . . . . 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 49673	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
12	2	onsetreclem3 49689	. . . . 5 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))
13		ssel 3937	. . . . 5 ⊢ ((𝐹‘𝑎) ⊆ setrecs(𝐹) → (𝑎 ∈ (𝐹‘𝑎) → 𝑎 ∈ setrecs(𝐹)))
14	11, 12, 13	syl2im 40	. . . 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 3046	. 2 ⊢ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
17		tfi 7809	. 2 ⊢ ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On)
18	7, 16, 17	mp2an 692	1 ⊢ setrecs(𝐹) = On

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ⊆ wss 3911  {cpr 4587  ∪ cuni 4867   ↦ cmpt 5183  Oncon0 6320  suc csuc 6322  ‘cfv 6499  setrecscsetrecs 49665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-r1 9693  df-rank 9694  df-setrecs 49666

This theorem is referenced by: (None)