Theorem mh-inf3sn 36783

Description: Version of inf3 9551 for the set of Zermelo ordinals ∅, {∅}, {{∅}}, {{{∅}}}, etc., where the successor of 𝑦 is {𝑦}. Unlike inf3 9551, the proof does not require ax-reg 9501, since the singleton properties snnz 4710 and sneqr 4773 are sufficient to guarantee that all elements of the sequence are distinct. (Contributed by Matthew House, 13-Apr-2026.)

Hypothesis

Ref	Expression
mh-inf3sn.1	⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)

Assertion

Ref	Expression
mh-inf3sn	⊢ ω ∈ V

Distinct variable group:   𝑥,𝑦

Proof of Theorem mh-inf3sn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 486	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
2		vex 3437	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	sneqr 4773	. . . . . 6 ⊢ ({𝑦} = {𝑧} → 𝑦 = 𝑧)
4	3	rgen2w 3060	. . . . 5 ⊢ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ({𝑦} = {𝑧} → 𝑦 = 𝑧)
5		eqid 2741	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↦ {𝑦}) = (𝑦 ∈ 𝑥 ↦ {𝑦})
6		sneq 4567	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
7	5, 6	f1mpt 7208	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 ↦ {𝑦}):𝑥–1-1→𝑥 ↔ (∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ({𝑦} = {𝑧} → 𝑦 = 𝑧)))
8	1, 4, 7	sylanblrc 597	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (𝑦 ∈ 𝑥 ↦ {𝑦}):𝑥–1-1→𝑥)
9		simpl 484	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ 𝑥)
10		snnzg 4708	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 → {𝑦} ≠ ∅)
11	10	necomd 2991	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ∅ ≠ {𝑦})
12	11	neneqd 2941	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → ¬ ∅ = {𝑦})
13	12	nrex 3069	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ 𝑥 ∅ = {𝑦}
14		vsnex 5366	. . . . . . . 8 ⊢ {𝑦} ∈ V
15	5, 14	elrnmpti 5910	. . . . . . 7 ⊢ (∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦}) ↔ ∃𝑦 ∈ 𝑥 ∅ = {𝑦})
16	13, 15	mtbir 325	. . . . . 6 ⊢ ¬ ∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦})
17	16	a1i 11	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ¬ ∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦}))
18	9, 17	eldifd 3895	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ (𝑥 ∖ ran (𝑦 ∈ 𝑥 ↦ {𝑦})))
19	8, 18	mh-inf3f1 36782	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥)
20		vex 3437	. . 3 ⊢ 𝑥 ∈ V
21		f1dmex 7901	. . 3 ⊢ (((rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥 ∧ 𝑥 ∈ V) → ω ∈ V)
22	19, 20, 21	sylancl 593	. 2 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ω ∈ V)
23		mh-inf3sn.1	. 2 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
24	22, 23	exlimiiv 1939	1 ⊢ ω ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  Vcvv 3433  ∅c0 4263  {csn 4557   ↦ cmpt 5155  ran crn 5621   ↾ cres 5622  –1-1→wf1 6485  ωcom 7809  reccrdg 8342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-oadd 8403

This theorem is referenced by: (None)