Theorem mh-inf3sn 37019

Description: Version of inf3 9603 for the set of Zermelo ordinals ∅, {∅}, {{∅}}, {{{∅}}}, etc., where the successor of 𝑦 is {𝑦}. Unlike inf3 9603, the proof does not require ax-reg 9553, since the singleton properties snnz 4741 and sneqr 4804 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 489	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
2		vex 3457	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	sneqr 4804	. . . . . 6 ⊢ ({𝑦} = {𝑧} → 𝑦 = 𝑧)
4	3	rgen2w 3082	. . . . 5 ⊢ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ({𝑦} = {𝑧} → 𝑦 = 𝑧)
5		eqid 2761	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↦ {𝑦}) = (𝑦 ∈ 𝑥 ↦ {𝑦})
6		sneq 4598	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
7	5, 6	f1mpt 7259	. . . . 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 601	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (𝑦 ∈ 𝑥 ↦ {𝑦}):𝑥–1-1→𝑥)
9		simpl 487	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ 𝑥)
10		snnzg 4739	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 → {𝑦} ≠ ∅)
11	10	necomd 3011	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ∅ ≠ {𝑦})
12	11	neneqd 2961	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → ¬ ∅ = {𝑦})
13	12	nrex 3091	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ 𝑥 ∅ = {𝑦}
14		vsnex 5406	. . . . . . . 8 ⊢ {𝑦} ∈ V
15	5, 14	elrnmpti 5952	. . . . . . 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 3915	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ (𝑥 ∖ ran (𝑦 ∈ 𝑥 ↦ {𝑦})))
19	8, 18	mh-inf3f1 37018	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥)
20		vex 3457	. . 3 ⊢ 𝑥 ∈ V
21		f1dmex 7953	. . 3 ⊢ (((rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥 ∧ 𝑥 ∈ V) → ω ∈ V)
22	19, 20, 21	sylancl 597	. 2 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ω ∈ V)
23		mh-inf3sn.1	. 2 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
24	22, 23	exlimiiv 1959	1 ⊢ ω ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087  Vcvv 3453  ∅c0 4285  {csn 4588   ↦ cmpt 5191  ran crn 5662   ↾ cres 5663  –1-1→wf1 6533  ωcom 7861  reccrdg 8395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8456

This theorem is referenced by: (None)