cnfin0 - Mathbox for ML

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Theorem cnfin0 33577

Description: The empty set is an ordinal in Cantor normal form. (Contributed by ML, 24-Jun-2022.)

**Hypotheses**
Ref	Expression
cnfin.1	⊢ 𝐼 = {⟨∅, 1_𝑜⟩}
cnfin.add	⊢ + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦‘𝑛) +_𝑜 (𝑧‘𝑛))))
cnfin.tr	⊢ (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))
cnfin.ltadd	⊢ (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))
cnfin.ltexp	⊢ (𝜒 ↔ ∃𝑎∃𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1_𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1_𝑜⟩})))
cnfin.yrule	⊢ 𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓 ∨ 𝜒)))}
cnfin.lt	⊢ < = ∪ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
cnfin.def	⊢ 𝐶 = dom <

**Assertion**
Ref	Expression
cnfin0	⊢ ∅ ∈ 𝐶

**Proof of Theorem cnfin0**
Step	Hyp	Ref	Expression
1		opex 5060	. . . . . 6 ⊢ ⟨∅, 𝐼⟩ ∈ V
2	1	snid 4347	. . . . 5 ⊢ ⟨∅, 𝐼⟩ ∈ {⟨∅, 𝐼⟩}
3		snex 5036	. . . . . . 7 ⊢ {⟨∅, 𝐼⟩} ∈ V
4		fr0g 7684	. . . . . . 7 ⊢ ({⟨∅, 𝐼⟩} ∈ V → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅) = {⟨∅, 𝐼⟩})
5	3, 4	ax-mp 5	. . . . . 6 ⊢ ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅) = {⟨∅, 𝐼⟩}
6		frfnom 7683	. . . . . . 7 ⊢ (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) Fn ω
7		peano1 7232	. . . . . . 7 ⊢ ∅ ∈ ω
8		fnfvelrn 6499	. . . . . . 7 ⊢ (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅) ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω))
9	6, 7, 8	mp2an 672	. . . . . 6 ⊢ ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅) ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
10	5, 9	eqeltrri 2847	. . . . 5 ⊢ {⟨∅, 𝐼⟩} ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
11		elunii 4579	. . . . 5 ⊢ ((⟨∅, 𝐼⟩ ∈ {⟨∅, 𝐼⟩} ∧ {⟨∅, 𝐼⟩} ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)) → ⟨∅, 𝐼⟩ ∈ ∪ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω))
12	2, 10, 11	mp2an 672	. . . 4 ⊢ ⟨∅, 𝐼⟩ ∈ ∪ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
13		cnfin.lt	. . . 4 ⊢ < = ∪ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
14	12, 13	eleqtrri 2849	. . 3 ⊢ ⟨∅, 𝐼⟩ ∈ <
15		0ex 4924	. . . 4 ⊢ ∅ ∈ V
16		cnfin.1	. . . . 5 ⊢ 𝐼 = {⟨∅, 1_𝑜⟩}
17		snex 5036	. . . . 5 ⊢ {⟨∅, 1_𝑜⟩} ∈ V
18	16, 17	eqeltri 2846	. . . 4 ⊢ 𝐼 ∈ V
19	15, 18	opeldm 5466	. . 3 ⊢ (⟨∅, 𝐼⟩ ∈ < → ∅ ∈ dom < )
20	14, 19	ax-mp 5	. 2 ⊢ ∅ ∈ dom <
21		cnfin.def	. 2 ⊢ 𝐶 = dom <
22	20, 21	eleqtrri 2849	1 ⊢ ∅ ∈ 𝐶

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 382 ∨ wo 836 = wceq 1631 ∃wex 1852 ∈ wcel 2145 ∃wrex 3062 Vcvv 3351 ∪ cun 3721 ∅c0 4063 {csn 4316 ⟨cop 4322 ∪ cuni 4574 {copab 4846 ↦ cmpt 4863 dom cdm 5249 ran crn 5250 ↾ cres 5251 Fn wfn 6026 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 ωcom 7212 reccrdg 7658 1_𝑜c1o 7706 +_𝑜 coa 7710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659

This theorem is referenced by: (None)

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