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Theorem cdadom1 9345
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 5142 . . . . 5 {∅} ∈ V
21xpdom1 8349 . . . 4 (𝐴𝐵 → (𝐴 × {∅}) ≼ (𝐵 × {∅}))
3 snex 5142 . . . . . 6 {1o} ∈ V
4 xpexg 7239 . . . . . 6 ((𝐶 ∈ V ∧ {1o} ∈ V) → (𝐶 × {1o}) ∈ V)
53, 4mpan2 681 . . . . 5 (𝐶 ∈ V → (𝐶 × {1o}) ∈ V)
6 domrefg 8278 . . . . 5 ((𝐶 × {1o}) ∈ V → (𝐶 × {1o}) ≼ (𝐶 × {1o}))
75, 6syl 17 . . . 4 (𝐶 ∈ V → (𝐶 × {1o}) ≼ (𝐶 × {1o}))
8 xp01disj 7862 . . . . 5 ((𝐵 × {∅}) ∩ (𝐶 × {1o})) = ∅
9 undom 8338 . . . . 5 ((((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1o}) ≼ (𝐶 × {1o})) ∧ ((𝐵 × {∅}) ∩ (𝐶 × {1o})) = ∅) → ((𝐴 × {∅}) ∪ (𝐶 × {1o})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1o})))
108, 9mpan2 681 . . . 4 (((𝐴 × {∅}) ≼ (𝐵 × {∅}) ∧ (𝐶 × {1o}) ≼ (𝐶 × {1o})) → ((𝐴 × {∅}) ∪ (𝐶 × {1o})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1o})))
112, 7, 10syl2an 589 . . 3 ((𝐴𝐵𝐶 ∈ V) → ((𝐴 × {∅}) ∪ (𝐶 × {1o})) ≼ ((𝐵 × {∅}) ∪ (𝐶 × {1o})))
12 reldom 8249 . . . . 5 Rel ≼
1312brrelex1i 5408 . . . 4 (𝐴𝐵𝐴 ∈ V)
14 cdaval 9329 . . . 4 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1o})))
1513, 14sylan 575 . . 3 ((𝐴𝐵𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1o})))
1612brrelex2i 5409 . . . 4 (𝐴𝐵𝐵 ∈ V)
17 cdaval 9329 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1o})))
1816, 17sylan 575 . . 3 ((𝐴𝐵𝐶 ∈ V) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1o})))
1911, 15, 183brtr4d 4920 . 2 ((𝐴𝐵𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
20 simpr 479 . . . . 5 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
2120intnand 484 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → ¬ (𝐴 ∈ V ∧ 𝐶 ∈ V))
22 cdafn 9328 . . . . . 6 +𝑐 Fn (V × V)
23 fndm 6237 . . . . . 6 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
2422, 23ax-mp 5 . . . . 5 dom +𝑐 = (V × V)
2524ndmov 7097 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
2621, 25syl 17 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ∅)
27 ovex 6956 . . . 4 (𝐵 +𝑐 𝐶) ∈ V
28270dom 8380 . . 3 ∅ ≼ (𝐵 +𝑐 𝐶)
2926, 28syl6eqbr 4927 . 2 ((𝐴𝐵 ∧ ¬ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
3019, 29pm2.61dan 803 1 (𝐴𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  cun 3790  cin 3791  c0 4141  {csn 4398   class class class wbr 4888   × cxp 5355  dom cdm 5357   Fn wfn 6132  (class class class)co 6924  1oc1o 7838  cdom 8241   +𝑐 ccda 9326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-1o 7845  df-en 8244  df-dom 8245  df-cda 9327
This theorem is referenced by:  cdadom2  9346  cdalepw  9355  unctb  9364  infdif  9368  gchcdaidm  9827  gchpwdom  9829  gchhar  9838
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