Theorem nulge 4457

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 4453	. . . . 5 ⊢ (A +c ∅) = ∅
2	1	eqcomi 2357	. . . 4 ⊢ ∅ = (A +c ∅)
3		addceq2 4385	. . . . . 6 ⊢ (x = ∅ → (A +c x) = (A +c ∅))
4	3	eqeq2d 2364	. . . . 5 ⊢ (x = ∅ → (∅ = (A +c x) ↔ ∅ = (A +c ∅)))
5	4	rspcev 2956	. . . 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 4449	. . 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 2616  ∅c0 3551  ⟪copk 4058   Nn cnnc 4374   +_c cplc 4376   ≤_fin clefin 4433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194  df-addc 4379  df-lefin 4441

This theorem is referenced by:  lenltfin  4470