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 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 (𝐴𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))

StepHypRef Expression
1 snex 4899 . . . . 5 {∅} ∈ V
21xpdom1 8044 . . . 4 (𝐴𝐵 → (𝐴 × {∅}) ≼ (𝐵 × {∅}))
3 snex 4899 . . . . . 6 {1𝑜} ∈ V
4 xpexg 6945 . . . . . 6 ((𝐶 ∈ V ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
53, 4mpan2 706 . . . . 5 (𝐶 ∈ V → (𝐶 × {1𝑜}) ∈ V)
6 domrefg 7975 . . . . 5 ((𝐶 × {1𝑜}) ∈ V → (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜}))
75, 6syl 17 . . . 4 (𝐶 ∈ V → (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜}))
8 xp01disj 7561 . . . . 5 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
9 undom 8033 . . . . 5 ((((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜})) ∧ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
108, 9mpan2 706 . . . 4 (((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
112, 7, 10syl2an 494 . . 3 ((𝐴𝐵𝐶 ∈ V) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
12 reldom 7946 . . . . 5 Rel ≼
1312brrelexi 5148 . . . 4 (𝐴𝐵𝐴 ∈ V)
14 cdaval 8977 . . . 4 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
1513, 14sylan 488 . . 3 ((𝐴𝐵𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
1612brrelex2i 5149 . . . 4 (𝐴𝐵𝐵 ∈ V)
17 cdaval 8977 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
1816, 17sylan 488 . . 3 ((𝐴𝐵𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
1911, 15, 183brtr4d 4676 . 2 ((𝐴𝐵𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
20 simpr 477 . . . . 5 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
2120intnand 961 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ (𝐴 ∈ V ∧ 𝐶 ∈ V))
22 cdafn 8976 . . . . . 6 +𝑐 Fn (V × V)
23 fndm 5978 . . . . . 6 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
2422, 23ax-mp 5 . . . . 5 dom +𝑐 = (V × V)
2524ndmov 6803 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
2621, 25syl 17 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
27 ovex 6663 . . . 4 (𝐵 +𝑐 𝐶) ∈ V
28270dom 8075 . . 3 ∅ ≼ (𝐵 +𝑐 𝐶)
2926, 28syl6eqbr 4683 . 2 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
3019, 29pm2.61dan 831 1 (𝐴𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  Vcvv 3195   ∪ cun 3565   ∩ cin 3566  ∅c0 3907  {csn 4168   class class class wbr 4644   × cxp 5102  dom cdm 5104   Fn wfn 5871  (class class class)co 6635  1𝑜c1o 7538   ≼ cdom 7938   +𝑐 ccda 8974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-1o 7545  df-en 7941  df-dom 7942  df-cda 8975 This theorem is referenced by:  cdadom2  8994  cdalepw  9003  unctb  9012  infdif  9016  gchcdaidm  9475  gchpwdom  9477  gchhar  9486
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