Theorem cdadom1 9008

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 4908	. . . . 5 ⊢ {∅} ∈ V
2	1	xpdom1 8059	. . . 4 ⊢ (𝐴 ≼ 𝐵 → (𝐴 × {∅}) ≼ (𝐵 × {∅}))
3		snex 4908	. . . . . 6 ⊢ {1𝑜} ∈ V
4		xpexg 6960	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
5	3, 4	mpan2 707	. . . . 5 ⊢ (𝐶 ∈ V → (𝐶 × {1𝑜}) ∈ V)
6		domrefg 7990	. . . . 5 ⊢ ((𝐶 × {1𝑜}) ∈ V → (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜}))
7	5, 6	syl 17	. . . 4 ⊢ (𝐶 ∈ V → (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜}))
8		xp01disj 7576	. . . . 5 ⊢ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
9		undom 8048	. . . . 5 ⊢ ((((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜})) ∧ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
10	8, 9	mpan2 707	. . . 4 ⊢ (((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
11	2, 7, 10	syl2an 494	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
12		reldom 7961	. . . . 5 ⊢ Rel ≼
13	12	brrelexi 5158	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V)
14		cdaval 8992	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
15	13, 14	sylan 488	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
16	12	brrelex2i 5159	. . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
17		cdaval 8992	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
18	16, 17	sylan 488	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
19	11, 15, 18	3brtr4d 4685	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
20		simpr 477	. . . . 5 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
21	20	intnand 962	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ (𝐴 ∈ V ∧ 𝐶 ∈ V))
22		cdafn 8991	. . . . . 6 ⊢ +𝑐 Fn (V × V)
23		fndm 5990	. . . . . 6 ⊢ ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
24	22, 23	ax-mp 5	. . . . 5 ⊢ dom +𝑐 = (V × V)
25	24	ndmov 6818	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
26	21, 25	syl 17	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
27		ovex 6678	. . . 4 ⊢ (𝐵 +𝑐 𝐶) ∈ V
28	27	0dom 8090	. . 3 ⊢ ∅ ≼ (𝐵 +𝑐 𝐶)
29	26, 28	syl6eqbr 4692	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
30	19, 29	pm2.61dan 832	1 ⊢ (𝐴 ≼ 𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  dom cdm 5114   Fn wfn 5883  (class class class)co 6650  1_𝑜c1o 7553   ≼ cdom 7953   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-en 7956  df-dom 7957  df-cda 8990

This theorem is referenced by:  cdadom2  9009  cdalepw  9018  unctb  9027  infdif  9031  gchcdaidm  9490  gchpwdom  9492  gchhar  9501