Theorem mh-inf3sn 37372

Description: Version of inf3 9636 for the set of Zermelo ordinals ∅, {∅}, {{∅}}, {{{∅}}}, etc., where the successor of 𝑦 is {𝑦}. Unlike inf3 9636, the proof does not require ax-reg 9586, since the singleton properties snnz 4736 and sneqr 4799 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 490	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
2		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	sneqr 4799	. . . . . 6 ⊢ ({𝑦} = {𝑧} → 𝑦 = 𝑧)
4	3	rgen2w 3081	. . . . 5 ⊢ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ({𝑦} = {𝑧} → 𝑦 = 𝑧)
5		eqid 2760	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↦ {𝑦}) = (𝑦 ∈ 𝑥 ↦ {𝑦})
6		sneq 4593	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
7	5, 6	f1mpt 7265	. . . . 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 602	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (𝑦 ∈ 𝑥 ↦ {𝑦}):𝑥–1-1→𝑥)
9		simpl 488	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ 𝑥)
10		snnzg 4734	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 → {𝑦} ≠ ∅)
11	10	necomd 3010	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ∅ ≠ {𝑦})
12	11	neneqd 2960	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → ¬ ∅ = {𝑦})
13	12	nrex 3090	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ 𝑥 ∅ = {𝑦}
14		vsnex 5392	. . . . . . . 8 ⊢ {𝑦} ∈ V
15	5, 14	elrnmpti 5943	. . . . . . 7 ⊢ (∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦}) ↔ ∃𝑦 ∈ 𝑥 ∅ = {𝑦})
16	13, 15	mtbir 326	. . . . . 6 ⊢ ¬ ∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦})
17	16	a1i 11	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ¬ ∅ ∈ ran (𝑦 ∈ 𝑥 ↦ {𝑦}))
18	9, 17	eldifd 4050	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ (𝑥 ∖ ran (𝑦 ∈ 𝑥 ↦ {𝑦})))
19	8, 18	mh-inf3f1 37371	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥)
20		vex 3454	. . 3 ⊢ 𝑥 ∈ V
21		f1dmex 7969	. . 3 ⊢ (((rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥 ∧ 𝑥 ∈ V) → ω ∈ V)
22	19, 20, 21	sylancl 598	. 2 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ω ∈ V)
23		mh-inf3sn.1	. 2 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
24	22, 23	exlimiiv 1963	1 ⊢ ω ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450  ∅c0 4278  {csn 4583   ↦ cmpt 5185  ran crn 5651   ↾ cres 5652  –1-1→wf1 6535  ωcom 7877  reccrdg 8417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418

This theorem is used by: (None)