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Theorem nulge 4456

Description: If the empty set is a finite cardinal, then it is a maximal element. (Contributed by SF, 19-Jan-2015.)

**Assertion**
Ref	Expression
nulge	⊢ ((∅ ∈ Nn ∧ A ∈ V) → ⟪A, ∅⟫ ∈ ≤_fin )

Proof of Theorem nulge Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcnul1 4452	. . . . 5 ⊢ (A +_c ∅) = ∅
2	1	eqcomi 2357	. . . 4 ⊢ ∅ = (A +_c ∅)
3		addceq2 4384	. . . . . 6 ⊢ (x = ∅ → (A +_c x) = (A +_c ∅))
4	3	eqeq2d 2364	. . . . 5 ⊢ (x = ∅ → (∅ = (A +_c x) ↔ ∅ = (A +_c ∅)))
5	4	rspcev 2955	. . . 4 ⊢ ((∅ ∈ Nn ∧ ∅ = (A +_c ∅)) → ∃x ∈ Nn ∅ = (A +_c x))
6	2, 5	mpan2 652	. . 3 ⊢ (∅ ∈ Nn → ∃x ∈ Nn ∅ = (A +_c x))
7	6	adantr 451	. 2 ⊢ ((∅ ∈ Nn ∧ A ∈ V) → ∃x ∈ Nn ∅ = (A +_c x))
8		opklefing 4448	. . 3 ⊢ ((A ∈ V ∧ ∅ ∈ Nn ) → (⟪A, ∅⟫ ∈ ≤_fin ↔ ∃x ∈ Nn ∅ = (A +_c x)))
9	8	ancoms 439	. 2 ⊢ ((∅ ∈ Nn ∧ A ∈ V) → (⟪A, ∅⟫ ∈ ≤_fin ↔ ∃x ∈ Nn ∅ = (A +_c x)))
10	7, 9	mpbird 223	1 ⊢ ((∅ ∈ Nn ∧ A ∈ V) → ⟪A, ∅⟫ ∈ ≤_fin )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∅c0 3550 ⟪copk 4057 Nn cnnc 4373 +_c cplc 4375 ≤_fin clefin 4432

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-sik 4192 df-ssetk 4193 df-addc 4378 df-lefin 4440

This theorem is referenced by: lenltfin 4469