Theorem mh-inf3sn 36724

Description: Version of inf3 9556 for the set of Zermelo ordinals ∅, {∅}, {{∅}}, {{{∅}}}, etc., where the successor of 𝑦 is {𝑦}. Unlike inf3 9556, the proof does not require ax-reg 9507, since the singleton properties snnz 4720 and sneqr 4783 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 484	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
2		vex 3433	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	sneqr 4783	. . . . . 6 ⊢ ({𝑦} = {𝑧} → 𝑦 = 𝑧)
4	3	rgen2w 3056	. . . . 5 ⊢ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ({𝑦} = {𝑧} → 𝑦 = 𝑧)
5		eqid 2736	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↦ {𝑦}) = (𝑦 ∈ 𝑥 ↦ {𝑦})
6		sneq 4577	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
7	5, 6	f1mpt 7216	. . . . 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 591	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (𝑦 ∈ 𝑥 ↦ {𝑦}):𝑥–1-1→𝑥)
9		simpl 482	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ 𝑥)
10		snnzg 4718	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 → {𝑦} ≠ ∅)
11	10	necomd 2987	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ∅ ≠ {𝑦})
12	11	neneqd 2937	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → ¬ ∅ = {𝑦})
13	12	nrex 3065	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ 𝑥 ∅ = {𝑦}
14		vsnex 5377	. . . . . . . 8 ⊢ {𝑦} ∈ V
15	5, 14	elrnmpti 5917	. . . . . . 7 ⊢ (∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦}) ↔ ∃𝑦 ∈ 𝑥 ∅ = {𝑦})
16	13, 15	mtbir 323	. . . . . 6 ⊢ ¬ ∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦})
17	16	a1i 11	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ¬ ∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦}))
18	9, 17	eldifd 3900	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ (𝑥 ∖ ran (𝑦 ∈ 𝑥 ↦ {𝑦})))
19	8, 18	mh-inf3f1 36723	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥)
20		vex 3433	. . 3 ⊢ 𝑥 ∈ V
21		f1dmex 7910	. . 3 ⊢ (((rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥 ∧ 𝑥 ∈ V) → ω ∈ V)
22	19, 20, 21	sylancl 587	. 2 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ω ∈ V)
23		mh-inf3sn.1	. 2 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
24	22, 23	exlimiiv 1933	1 ⊢ ω ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429  ∅c0 4273  {csn 4567   ↦ cmpt 5166  ran crn 5632   ↾ cres 5633  –1-1→wf1 6495  ωcom 7817  reccrdg 8348

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-pr 5375  ax-un 7689

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-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-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409

This theorem is referenced by: (None)