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Theorem cdadom1 9200
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))

Proof of Theorem cdadom1
StepHypRef Expression
1 snex 5057 . . . . 5 {∅} ∈ V
21xpdom1 8224 . . . 4 (𝐴𝐵 → (𝐴 × {∅}) ≼ (𝐵 × {∅}))
3 snex 5057 . . . . . 6 {1𝑜} ∈ V
4 xpexg 7125 . . . . . 6 ((𝐶 ∈ V ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
53, 4mpan2 709 . . . . 5 (𝐶 ∈ V → (𝐶 × {1𝑜}) ∈ V)
6 domrefg 8156 . . . . 5 ((𝐶 × {1𝑜}) ∈ V → (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜}))
75, 6syl 17 . . . 4 (𝐶 ∈ V → (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜}))
8 xp01disj 7745 . . . . 5 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
9 undom 8213 . . . . 5 ((((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜})) ∧ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
108, 9mpan2 709 . . . 4 (((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≼ (𝐶 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
112, 7, 10syl2an 495 . . 3 ((𝐴𝐵𝐶 ∈ V) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
12 reldom 8127 . . . . 5 Rel ≼
1312brrelexi 5315 . . . 4 (𝐴𝐵𝐴 ∈ V)
14 cdaval 9184 . . . 4 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
1513, 14sylan 489 . . 3 ((𝐴𝐵𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
1612brrelex2i 5316 . . . 4 (𝐴𝐵𝐵 ∈ V)
17 cdaval 9184 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
1816, 17sylan 489 . . 3 ((𝐴𝐵𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
1911, 15, 183brtr4d 4836 . 2 ((𝐴𝐵𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
20 simpr 479 . . . . 5 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
2120intnand 1000 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ (𝐴 ∈ V ∧ 𝐶 ∈ V))
22 cdafn 9183 . . . . . 6 +𝑐 Fn (V × V)
23 fndm 6151 . . . . . 6 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
2422, 23ax-mp 5 . . . . 5 dom +𝑐 = (V × V)
2524ndmov 6983 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
2621, 25syl 17 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
27 ovex 6841 . . . 4 (𝐵 +𝑐 𝐶) ∈ V
28270dom 8255 . . 3 ∅ ≼ (𝐵 +𝑐 𝐶)
2926, 28syl6eqbr 4843 . 2 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
3019, 29pm2.61dan 867 1 (𝐴𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  cin 3714  c0 4058  {csn 4321   class class class wbr 4804   × cxp 5264  dom cdm 5266   Fn wfn 6044  (class class class)co 6813  1𝑜c1o 7722  cdom 8119   +𝑐 ccda 9181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-1o 7729  df-en 8122  df-dom 8123  df-cda 9182
This theorem is referenced by:  cdadom2  9201  cdalepw  9210  unctb  9219  infdif  9223  gchcdaidm  9682  gchpwdom  9684  gchhar  9693
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