Theorem onsetrec 50183

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 50181 and onsetreclem3 50182 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 35972.

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 2736	. . . 4 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		onsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
3	2	onsetreclem2 50181	. . . . . 6 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)
4	3	ax-gen 1797	. . . . 5 ⊢ ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)
5	4	a1i 11	. . . 4 ⊢ (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On))
6	1, 5	setrec2v 50171	. . 3 ⊢ (⊤ → setrecs(𝐹) ⊆ On)
7	6	mptru 1549	. 2 ⊢ setrecs(𝐹) ⊆ On
8		vex 3433	. . . . . . 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 50166	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
12	2	onsetreclem3 50182	. . . . 5 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))
13		ssel 3915	. . . . 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 3053	. 2 ⊢ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
17		tfi 7804	. 2 ⊢ ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On)
18	7, 16, 17	mp2an 693	1 ⊢ setrecs(𝐹) = On

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  {cpr 4569  ∪ cuni 4850   ↦ cmpt 5166  Oncon0 6323  suc csuc 6325  ‘cfv 6498  setrecscsetrecs 50158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-r1 9688  df-rank 9689  df-setrecs 50159

This theorem is referenced by: (None)