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 )

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