Theorem addcnul1 4453

Description: Cardinal addition with the empty set. Theorem X.1.20, corollary 1 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.)

Assertion

Ref	Expression
addcnul1	⊢ (A +c ∅) = ∅

Proof of Theorem addcnul1

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eq0 3565	. 2 ⊢ ((A +c ∅) = ∅ ↔ ∀a ¬ a ∈ (A +c ∅))
2		rex0 3564	. . . . 5 ⊢ ¬ ∃c ∈ ∅ ((b ∩ c) = ∅ ∧ a = (b ∪ c))
3	2	a1i 10	. . . 4 ⊢ (b ∈ A → ¬ ∃c ∈ ∅ ((b ∩ c) = ∅ ∧ a = (b ∪ c)))
4	3	nrex 2717	. . 3 ⊢ ¬ ∃b ∈ A ∃c ∈ ∅ ((b ∩ c) = ∅ ∧ a = (b ∪ c))
5		eladdc 4399	. . 3 ⊢ (a ∈ (A +c ∅) ↔ ∃b ∈ A ∃c ∈ ∅ ((b ∩ c) = ∅ ∧ a = (b ∪ c)))
6	4, 5	mtbir 290	. 2 ⊢ ¬ a ∈ (A +c ∅)
7	1, 6	mpgbir 1550	1 ⊢ (A +c ∅) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∪ cun 3208   ∩ cin 3209  ∅c0 3551   +_c cplc 4376

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nul 3552  df-addc 4379

This theorem is referenced by:  addcnnul  4454  nulge  4457  tfinltfinlem1  4501  eventfin  4518  oddtfin  4519