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Theorem addcnnul 4453

Description: If cardinal addition is non-empty, then both addends are non-empty. Theorem X.1.20 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.)

Assertion

Ref	Expression
addcnnul	⊢ ((A +c B) ≠ ∅ → (A ≠ ∅ ∧ B ≠ ∅))

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Proof of Theorem addcnnul

Step	Hyp	Ref	Expression
1		addceq1 4383	. . . 4 ⊢ (A = ∅ → (A +c B) = (∅ +c B))
2		addccom 4406	. . . . 5 ⊢ (∅ +c B) = (B +c ∅)
3		addcnul1 4452	. . . . 5 ⊢ (B +c ∅) = ∅
4	2, 3	eqtri 2373	. . . 4 ⊢ (∅ +c B) = ∅
5	1, 4	syl6eq 2401	. . 3 ⊢ (A = ∅ → (A +c B) = ∅)
6	5	necon3i 2555	. 2 ⊢ ((A +c B) ≠ ∅ → A ≠ ∅)
7		addceq2 4384	. . . 4 ⊢ (B = ∅ → (A +c B) = (A +c ∅))
8		addcnul1 4452	. . . 4 ⊢ (A +c ∅) = ∅
9	7, 8	syl6eq 2401	. . 3 ⊢ (B = ∅ → (A +c B) = ∅)
10	9	necon3i 2555	. 2 ⊢ ((A +c B) ≠ ∅ → B ≠ ∅)
11	6, 10	jca 518	1 ⊢ ((A +c B) ≠ ∅ → (A ≠ ∅ ∧ B ≠ ∅))

Colors of variables: wff setvar class

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516  ∅c0 3550   +c cplc 4375

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

This theorem is referenced by:  preaddccan2  4455  leltfintr  4458  ltfintri  4466  tfinltfinlem1  4500  evenoddnnnul  4514  evenodddisj  4516  oddtfin  4518

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