Theorem addccom 4407

Description: Cardinal sum commutes. Theorem X.1.9 of [Rosser] p. 276. (Contributed by SF, 15-Jan-2015.)

Assertion

Ref	Expression
addccom	⊢ (A +c B) = (B +c A)

Proof of Theorem addccom

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		incom 3449	. . . . . . 7 ⊢ (y ∩ z) = (z ∩ y)
2	1	eqeq1i 2360	. . . . . 6 ⊢ ((y ∩ z) = ∅ ↔ (z ∩ y) = ∅)
3		uncom 3409	. . . . . . 7 ⊢ (y ∪ z) = (z ∪ y)
4	3	eqeq2i 2363	. . . . . 6 ⊢ (x = (y ∪ z) ↔ x = (z ∪ y))
5	2, 4	anbi12i 678	. . . . 5 ⊢ (((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ ((z ∩ y) = ∅ ∧ x = (z ∪ y)))
6	5	2rexbii 2642	. . . 4 ⊢ (∃y ∈ A ∃z ∈ B ((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ ∃y ∈ A ∃z ∈ B ((z ∩ y) = ∅ ∧ x = (z ∪ y)))
7		rexcom 2773	. . . 4 ⊢ (∃y ∈ A ∃z ∈ B ((z ∩ y) = ∅ ∧ x = (z ∪ y)) ↔ ∃z ∈ B ∃y ∈ A ((z ∩ y) = ∅ ∧ x = (z ∪ y)))
8	6, 7	bitri 240	. . 3 ⊢ (∃y ∈ A ∃z ∈ B ((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ ∃z ∈ B ∃y ∈ A ((z ∩ y) = ∅ ∧ x = (z ∪ y)))
9	8	abbii 2466	. 2 ⊢ {x ∣ ∃y ∈ A ∃z ∈ B ((y ∩ z) = ∅ ∧ x = (y ∪ z))} = {x ∣ ∃z ∈ B ∃y ∈ A ((z ∩ y) = ∅ ∧ x = (z ∪ y))}
10		df-addc 4379	. 2 ⊢ (A +c B) = {x ∣ ∃y ∈ A ∃z ∈ B ((y ∩ z) = ∅ ∧ x = (y ∪ z))}
11		df-addc 4379	. 2 ⊢ (B +c A) = {x ∣ ∃z ∈ B ∃y ∈ A ((z ∩ y) = ∅ ∧ x = (z ∪ y))}
12	9, 10, 11	3eqtr4i 2383	1 ⊢ (A +c B) = (B +c A)

Syntax hints:   ∧ wa 358   = wceq 1642  {cab 2339  ∃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

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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-addc 4379

This theorem is referenced by:  addcid2  4408  1cnnc  4409  addc32  4417  nncaddccl  4420  addcnnul  4454  ltfintr  4460  tfinltfinlem1  4501  oddtfin  4519  addccan1  4561  leaddc2  6216  nc0suc  6218  nmembers1lem3  6271  nchoicelem1  6290  nchoicelem7  6296  nchoicelem14  6303  nchoicelem17  6306