Theorem cdadom1 8891

Description: Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
cdadom1	⊢ (𝐴 ≼ 𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))

Proof of Theorem cdadom1

Step	Hyp	Ref	Expression
1		snex 4835	. . . . 5 ⊢ {∅} ∈ V
2	1	xpdom1 7944	. . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝐴 × {∅}) ≼ (𝐵 × {∅}))
3		snex 4835	. . . . . 6 ⊢ {1𝑜} ∈ V
4		xpexg 6858	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
5	3, 4	mpan2 703	. . . . 5 ⊢ (𝐶 ∈ V → (𝐶 × {1𝑜}) ∈ V)
6		domrefg 7876	. . . . 5 ⊢ ((𝐶 × {1𝑜}) ∈ V → (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜}))
7	5, 6	syl 17	. . . 4 ⊢ (𝐶 ∈ V → (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜}))
8		xp01disj 7463	. . . . 5 ⊢ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
9		undom 7933	. . . . 5 ⊢ ((((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜})) ∧ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
10	8, 9	mpan2 703	. . . 4 ⊢ (((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
11	2, 7, 10	syl2an 493	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
12		reldom 7847	. . . . 5 ⊢ Rel ≼
13	12	brrelexi 5082	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
14		cdaval 8875	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
15	13, 14	sylan 487	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
16	12	brrelex2i 5083	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
17		cdaval 8875	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
18	16, 17	sylan 487	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
19	11, 15, 18	3brtr4d 4615	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
20		simpr 476	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
21	20	intnand 953	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ (𝐴 ∈ V ∧ 𝐶 ∈ V))
22		cdafn 8874	. . . . . 6 ⊢ +𝑐 Fn (V × V)
23		fndm 5904	. . . . . 6 ⊢ ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
24	22, 23	ax-mp 5	. . . . 5 ⊢ dom +𝑐 = (V × V)
25	24	ndmov 6716	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
26	21, 25	syl 17	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
27		ovex 6577	. . . 4 ⊢ (𝐵 +𝑐 𝐶) ∈ V
28	27	0dom 7975	. . 3 ⊢ ∅ ≼ (𝐵 +𝑐 𝐶)
29	26, 28	syl6eqbr 4622	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
30	19, 29	pm2.61dan 828	1 ⊢ (𝐴 ≼ 𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  dom cdm 5038   Fn wfn 5799  (class class class)co 6549  1_𝑜c1o 7440   ≼ cdom 7839   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-en 7842  df-dom 7843  df-cda 8873

This theorem is referenced by:  cdadom2  8892  cdalepw  8901  unctb  8910  infdif  8914  gchcdaidm  9369  gchpwdom  9371  gchhar  9380