Theorem onsetrec 50198

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 50196 and onsetreclem3 50197 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 35991.

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 2737	. . . 4 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		onsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
3	2	onsetreclem2 50196	. . . . . 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 50186	. . 3 ⊢ (⊤ → setrecs(𝐹) ⊆ On)
7	6	mptru 1549	. 2 ⊢ setrecs(𝐹) ⊆ On
8		vex 3434	. . . . . . 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 50181	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
12	2	onsetreclem3 50197	. . . . 5 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))
13		ssel 3916	. . . . 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 3054	. 2 ⊢ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
17		tfi 7798	. 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 3052  Vcvv 3430   ⊆ wss 3890  {cpr 4570  ∪ cuni 4851   ↦ cmpt 5167  Oncon0 6318  suc csuc 6320  ‘cfv 6493  setrecscsetrecs 50173

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-reg 9501  ax-inf2 9556

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9682  df-rank 9683  df-setrecs 50174

This theorem is referenced by: (None)