Theorem mh-inf3sn 36907

Description: Version of inf3 9592 for the set of Zermelo ordinals ∅, {∅}, {{∅}}, {{{∅}}}, etc., where the successor of 𝑦 is {𝑦}. Unlike inf3 9592, the proof does not require ax-reg 9542, since the singleton properties snnz 4737 and sneqr 4800 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 488	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
2		vex 3460	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	sneqr 4800	. . . . . 6 ⊢ ({𝑦} = {𝑧} → 𝑦 = 𝑧)
4	3	rgen2w 3083	. . . . 5 ⊢ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ({𝑦} = {𝑧} → 𝑦 = 𝑧)
5		eqid 2764	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↦ {𝑦}) = (𝑦 ∈ 𝑥 ↦ {𝑦})
6		sneq 4594	. . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
7	5, 6	f1mpt 7247	. . . . 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 599	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (𝑦 ∈ 𝑥 ↦ {𝑦}):𝑥–1-1→𝑥)
9		simpl 486	. . . . 5 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ 𝑥)
10		snnzg 4735	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 → {𝑦} ≠ ∅)
11	10	necomd 3014	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ∅ ≠ {𝑦})
12	11	neneqd 2964	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → ¬ ∅ = {𝑦})
13	12	nrex 3092	. . . . . . 7 ⊢ ¬ ∃𝑦 ∈ 𝑥 ∅ = {𝑦}
14		vsnex 5394	. . . . . . . 8 ⊢ {𝑦} ∈ V
15	5, 14	elrnmpti 5940	. . . . . . 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 3917	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ∅ ∈ (𝑥 ∖ ran (𝑦 ∈ 𝑥 ↦ {𝑦})))
19	8, 18	mh-inf3f1 36906	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → (rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥)
20		vex 3460	. . 3 ⊢ 𝑥 ∈ V
21		f1dmex 7940	. . 3 ⊢ (((rec((𝑦 ∈ 𝑥 ↦ {𝑦}), ∅) ↾ ω):ω–1-1→𝑥 ∧ 𝑥 ∈ V) → ω ∈ V)
22	19, 20, 21	sylancl 595	. 2 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) → ω ∈ V)
23		mh-inf3sn.1	. 2 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥)
24	22, 23	exlimiiv 1953	1 ⊢ ω ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088  Vcvv 3456  ∅c0 4287  {csn 4584   ↦ cmpt 5183  ran crn 5650   ↾ cres 5651  –1-1→wf1 6520  ωcom 7848  reccrdg 8382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-oadd 8443

This theorem is referenced by: (None)