New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ce0addcnnul GIF version

 Description: The sum of two cardinals raised to 0c is non-empty iff each addend raised to 0c is non-empty. Theorem XI.2.43 of [Rosser] p. 383. (Contributed by SF, 9-Mar-2015.)
Assertion
Ref Expression
ce0addcnnul ((M NC N NC ) → (((M +c N) ↑c 0c) ≠ ↔ ((Mc 0c) ≠ (Nc 0c) ≠ )))

Dummy variables a b g p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ncaddccl 6144 . . . 4 ((M NC N NC ) → (M +c N) NC )
2 ce0nnul 6177 . . . . 5 ((M +c N) NC → (((M +c N) ↑c 0c) ≠ a1a (M +c N)))
3 eladdc 4398 . . . . . 6 (1a (M +c N) ↔ b M g N ((bg) = 1a = (bg)))
43exbii 1582 . . . . 5 (a1a (M +c N) ↔ ab M g N ((bg) = 1a = (bg)))
52, 4syl6bb 252 . . . 4 ((M +c N) NC → (((M +c N) ↑c 0c) ≠ ab M g N ((bg) = 1a = (bg))))
61, 5syl 15 . . 3 ((M NC N NC ) → (((M +c N) ↑c 0c) ≠ ab M g N ((bg) = 1a = (bg))))
7 ncseqnc 6128 . . . . . . . 8 (M NC → (M = Nc bb M))
8 ncseqnc 6128 . . . . . . . 8 (N NC → (N = Nc gg N))
97, 8bi2anan9 843 . . . . . . 7 ((M NC N NC ) → ((M = Nc b N = Nc g) ↔ (b M g N)))
109biimpar 471 . . . . . 6 (((M NC N NC ) (b M g N)) → (M = Nc b N = Nc g))
11 ssun1 3426 . . . . . . . . . . . 12 b (bg)
12 id 19 . . . . . . . . . . . 12 (1a = (bg) → 1a = (bg))
1311, 12syl5sseqr 3320 . . . . . . . . . . 11 (1a = (bg) → b 1a)
14 ssun2 3427 . . . . . . . . . . . 12 g (bg)
1514, 12syl5sseqr 3320 . . . . . . . . . . 11 (1a = (bg) → g 1a)
1613, 15jca 518 . . . . . . . . . 10 (1a = (bg) → (b 1a g 1a))
17 vex 2862 . . . . . . . . . . . . 13 b V
1817sspw1 4335 . . . . . . . . . . . 12 (b 1ap(p a b = 1p))
19 vex 2862 . . . . . . . . . . . . 13 g V
2019sspw1 4335 . . . . . . . . . . . 12 (g 1aq(q a g = 1q))
2118, 20anbi12i 678 . . . . . . . . . . 11 ((b 1a g 1a) ↔ (p(p a b = 1p) q(q a g = 1q)))
22 eeanv 1913 . . . . . . . . . . 11 (pq((p a b = 1p) (q a g = 1q)) ↔ (p(p a b = 1p) q(q a g = 1q)))
2321, 22bitr4i 243 . . . . . . . . . 10 ((b 1a g 1a) ↔ pq((p a b = 1p) (q a g = 1q)))
2416, 23sylib 188 . . . . . . . . 9 (1a = (bg) → pq((p a b = 1p) (q a g = 1q)))
25 pw1eq 4143 . . . . . . . . . . . . . . . . 17 (a = p1a = 1p)
2625eleq1d 2419 . . . . . . . . . . . . . . . 16 (a = p → (1a Nc 1p1p Nc 1p))
27 vex 2862 . . . . . . . . . . . . . . . . . 18 p V
2827pw1ex 4303 . . . . . . . . . . . . . . . . 17 1p V
2928ncid 6123 . . . . . . . . . . . . . . . 16 1p Nc 1p
3026, 29speiv 2000 . . . . . . . . . . . . . . 15 a1a Nc 1p
31 ncelncs 6120 . . . . . . . . . . . . . . . 16 (1p V → Nc 1p NC )
32 ce0nnul 6177 . . . . . . . . . . . . . . . 16 ( Nc 1p NC → (( Nc 1pc 0c) ≠ a1a Nc 1p))
3328, 31, 32mp2b 9 . . . . . . . . . . . . . . 15 (( Nc 1pc 0c) ≠ a1a Nc 1p)
3430, 33mpbir 200 . . . . . . . . . . . . . 14 ( Nc 1pc 0c) ≠
35 pw1eq 4143 . . . . . . . . . . . . . . . . 17 (a = q1a = 1q)
3635eleq1d 2419 . . . . . . . . . . . . . . . 16 (a = q → (1a Nc 1q1q Nc 1q))
37 vex 2862 . . . . . . . . . . . . . . . . . 18 q V
3837pw1ex 4303 . . . . . . . . . . . . . . . . 17 1q V
3938ncid 6123 . . . . . . . . . . . . . . . 16 1q Nc 1q
4036, 39speiv 2000 . . . . . . . . . . . . . . 15 a1a Nc 1q
41 ncelncs 6120 . . . . . . . . . . . . . . . 16 (1q V → Nc 1q NC )
42 ce0nnul 6177 . . . . . . . . . . . . . . . 16 ( Nc 1q NC → (( Nc 1qc 0c) ≠ a1a Nc 1q))
4338, 41, 42mp2b 9 . . . . . . . . . . . . . . 15 (( Nc 1qc 0c) ≠ a1a Nc 1q)
4440, 43mpbir 200 . . . . . . . . . . . . . 14 ( Nc 1qc 0c) ≠
4534, 44pm3.2i 441 . . . . . . . . . . . . 13 (( Nc 1pc 0c) ≠ ( Nc 1qc 0c) ≠ )
46 nceq 6108 . . . . . . . . . . . . . . . 16 (b = 1pNc b = Nc 1p)
4746oveq1d 5537 . . . . . . . . . . . . . . 15 (b = 1p → ( Nc bc 0c) = ( Nc 1pc 0c))
4847neeq1d 2529 . . . . . . . . . . . . . 14 (b = 1p → (( Nc bc 0c) ≠ ↔ ( Nc 1pc 0c) ≠ ))
49 nceq 6108 . . . . . . . . . . . . . . . 16 (g = 1qNc g = Nc 1q)
5049oveq1d 5537 . . . . . . . . . . . . . . 15 (g = 1q → ( Nc gc 0c) = ( Nc 1qc 0c))
5150neeq1d 2529 . . . . . . . . . . . . . 14 (g = 1q → (( Nc gc 0c) ≠ ↔ ( Nc 1qc 0c) ≠ ))
5248, 51bi2anan9 843 . . . . . . . . . . . . 13 ((b = 1p g = 1q) → ((( Nc bc 0c) ≠ ( Nc gc 0c) ≠ ) ↔ (( Nc 1pc 0c) ≠ ( Nc 1qc 0c) ≠ )))
5345, 52mpbiri 224 . . . . . . . . . . . 12 ((b = 1p g = 1q) → (( Nc bc 0c) ≠ ( Nc gc 0c) ≠ ))
5453ad2ant2l 726 . . . . . . . . . . 11 (((p a b = 1p) (q a g = 1q)) → (( Nc bc 0c) ≠ ( Nc gc 0c) ≠ ))
5554a1d 22 . . . . . . . . . 10 (((p a b = 1p) (q a g = 1q)) → ((bg) = → (( Nc bc 0c) ≠ ( Nc gc 0c) ≠ )))
5655exlimivv 1635 . . . . . . . . 9 (pq((p a b = 1p) (q a g = 1q)) → ((bg) = → (( Nc bc 0c) ≠ ( Nc gc 0c) ≠ )))
5724, 56syl 15 . . . . . . . 8 (1a = (bg) → ((bg) = → (( Nc bc 0c) ≠ ( Nc gc 0c) ≠ )))
5857impcom 419 . . . . . . 7 (((bg) = 1a = (bg)) → (( Nc bc 0c) ≠ ( Nc gc 0c) ≠ ))
59 oveq1 5530 . . . . . . . . 9 (M = Nc b → (Mc 0c) = ( Nc bc 0c))
6059neeq1d 2529 . . . . . . . 8 (M = Nc b → ((Mc 0c) ≠ ↔ ( Nc bc 0c) ≠ ))
61 oveq1 5530 . . . . . . . . 9 (N = Nc g → (Nc 0c) = ( Nc gc 0c))
6261neeq1d 2529 . . . . . . . 8 (N = Nc g → ((Nc 0c) ≠ ↔ ( Nc gc 0c) ≠ ))
6360, 62bi2anan9 843 . . . . . . 7 ((M = Nc b N = Nc g) → (((Mc 0c) ≠ (Nc 0c) ≠ ) ↔ (( Nc bc 0c) ≠ ( Nc gc 0c) ≠ )))
6458, 63syl5ibr 212 . . . . . 6 ((M = Nc b N = Nc g) → (((bg) = 1a = (bg)) → ((Mc 0c) ≠ (Nc 0c) ≠ )))
6510, 64syl 15 . . . . 5 (((M NC N NC ) (b M g N)) → (((bg) = 1a = (bg)) → ((Mc 0c) ≠ (Nc 0c) ≠ )))
6665rexlimdvva 2745 . . . 4 ((M NC N NC ) → (b M g N ((bg) = 1a = (bg)) → ((Mc 0c) ≠ (Nc 0c) ≠ )))
6766exlimdv 1636 . . 3 ((M NC N NC ) → (ab M g N ((bg) = 1a = (bg)) → ((Mc 0c) ≠ (Nc 0c) ≠ )))
686, 67sylbid 206 . 2 ((M NC N NC ) → (((M +c N) ↑c 0c) ≠ → ((Mc 0c) ≠ (Nc 0c) ≠ )))
69 ce0nnul 6177 . . . . 5 (M NC → ((Mc 0c) ≠ b1b M))
70 ce0nnul 6177 . . . . 5 (N NC → ((Nc 0c) ≠ g1g N))
7169, 70bi2anan9 843 . . . 4 ((M NC N NC ) → (((Mc 0c) ≠ (Nc 0c) ≠ ) ↔ (b1b M g1g N)))
72 eeanv 1913 . . . 4 (bg(1b M 1g N) ↔ (b1b M g1g N))
7371, 72syl6bbr 254 . . 3 ((M NC N NC ) → (((Mc 0c) ≠ (Nc 0c) ≠ ) ↔ bg(1b M 1g N)))
74 ncseqnc 6128 . . . . . 6 (M NC → (M = Nc 1b1b M))
75 ncseqnc 6128 . . . . . 6 (N NC → (N = Nc 1g1g N))
7674, 75bi2anan9 843 . . . . 5 ((M NC N NC ) → ((M = Nc 1b N = Nc 1g) ↔ (1b M 1g N)))
77 vvex 4109 . . . . . . . . . . . 12 V V
7817, 77xpsnen 6049 . . . . . . . . . . 11 (b × {V}) ≈ b
79 enpw1 6062 . . . . . . . . . . 11 ((b × {V}) ≈ b1(b × {V}) ≈ 1b)
8078, 79mpbi 199 . . . . . . . . . 10 1(b × {V}) ≈ 1b
81 snex 4111 . . . . . . . . . . . . 13 {V} V
8217, 81xpex 5115 . . . . . . . . . . . 12 (b × {V}) V
8382pw1ex 4303 . . . . . . . . . . 11 1(b × {V}) V
8483eqnc 6127 . . . . . . . . . 10 ( Nc 1(b × {V}) = Nc 1b1(b × {V}) ≈ 1b)
8580, 84mpbir 200 . . . . . . . . 9 Nc 1(b × {V}) = Nc 1b
86 0ex 4110 . . . . . . . . . . . 12 V
8719, 86xpsnen 6049 . . . . . . . . . . 11 (g × {}) ≈ g
88 enpw1 6062 . . . . . . . . . . 11 ((g × {}) ≈ g1(g × {}) ≈ 1g)
8987, 88mpbi 199 . . . . . . . . . 10 1(g × {}) ≈ 1g
90 snex 4111 . . . . . . . . . . . . 13 {} V
9119, 90xpex 5115 . . . . . . . . . . . 12 (g × {}) V
9291pw1ex 4303 . . . . . . . . . . 11 1(g × {}) V
9392eqnc 6127 . . . . . . . . . 10 ( Nc 1(g × {}) = Nc 1g1(g × {}) ≈ 1g)
9489, 93mpbir 200 . . . . . . . . 9 Nc 1(g × {}) = Nc 1g
9585, 94addceq12i 4388 . . . . . . . 8 ( Nc 1(b × {V}) +c Nc 1(g × {})) = ( Nc 1b +c Nc 1g)
9695oveq1i 5533 . . . . . . 7 (( Nc 1(b × {V}) +c Nc 1(g × {})) ↑c 0c) = (( Nc 1b +c Nc 1g) ↑c 0c)
97 pw1un 4163 . . . . . . . . . 10 1((b × {V}) ∪ (g × {})) = (1(b × {V}) ∪ 1(g × {}))
9883ncid 6123 . . . . . . . . . . 11 1(b × {V}) Nc 1(b × {V})
9992ncid 6123 . . . . . . . . . . 11 1(g × {}) Nc 1(g × {})
100 vn0 3557 . . . . . . . . . . . . . 14 V ≠
10177, 100xpnedisj 5513 . . . . . . . . . . . . 13 ((b × {V}) ∩ (g × {})) =
102 pw1eq 4143 . . . . . . . . . . . . 13 (((b × {V}) ∩ (g × {})) = 1((b × {V}) ∩ (g × {})) = 1)
103101, 102ax-mp 8 . . . . . . . . . . . 12 1((b × {V}) ∩ (g × {})) = 1
104 pw1in 4164 . . . . . . . . . . . 12 1((b × {V}) ∩ (g × {})) = (1(b × {V}) ∩ 1(g × {}))
105 pw10 4161 . . . . . . . . . . . 12 1 =
106103, 104, 1053eqtr3i 2381 . . . . . . . . . . 11 (1(b × {V}) ∩ 1(g × {})) =
107 eladdci 4399 . . . . . . . . . . 11 ((1(b × {V}) Nc 1(b × {V}) 1(g × {}) Nc 1(g × {}) (1(b × {V}) ∩ 1(g × {})) = ) → (1(b × {V}) ∪ 1(g × {})) ( Nc 1(b × {V}) +c Nc 1(g × {})))
10898, 99, 106, 107mp3an 1277 . . . . . . . . . 10 (1(b × {V}) ∪ 1(g × {})) ( Nc 1(b × {V}) +c Nc 1(g × {}))
10997, 108eqeltri 2423 . . . . . . . . 9 1((b × {V}) ∪ (g × {})) ( Nc 1(b × {V}) +c Nc 1(g × {}))
11082, 91unex 4106 . . . . . . . . . 10 ((b × {V}) ∪ (g × {})) V
111 pw1eq 4143 . . . . . . . . . . 11 (a = ((b × {V}) ∪ (g × {})) → 1a = 1((b × {V}) ∪ (g × {})))
112111eleq1d 2419 . . . . . . . . . 10 (a = ((b × {V}) ∪ (g × {})) → (1a ( Nc 1(b × {V}) +c Nc 1(g × {})) ↔ 1((b × {V}) ∪ (g × {})) ( Nc 1(b × {V}) +c Nc 1(g × {}))))
113110, 112spcev 2946 . . . . . . . . 9 (1((b × {V}) ∪ (g × {})) ( Nc 1(b × {V}) +c Nc 1(g × {})) → a1a ( Nc 1(b × {V}) +c Nc 1(g × {})))
114109, 113ax-mp 8 . . . . . . . 8 a1a ( Nc 1(b × {V}) +c Nc 1(g × {}))
11583ncelncsi 6121 . . . . . . . . . 10 Nc 1(b × {V}) NC
11692ncelncsi 6121 . . . . . . . . . 10 Nc 1(g × {}) NC
117 ncaddccl 6144 . . . . . . . . . 10 (( Nc 1(b × {V}) NC Nc 1(g × {}) NC ) → ( Nc 1(b × {V}) +c Nc 1(g × {})) NC )
118115, 116, 117mp2an 653 . . . . . . . . 9 ( Nc 1(b × {V}) +c Nc 1(g × {})) NC
119 ce0nnul 6177 . . . . . . . . 9 (( Nc 1(b × {V}) +c Nc 1(g × {})) NC → ((( Nc 1(b × {V}) +c Nc 1(g × {})) ↑c 0c) ≠ a1a ( Nc 1(b × {V}) +c Nc 1(g × {}))))
120118, 119ax-mp 8 . . . . . . . 8 ((( Nc 1(b × {V}) +c Nc 1(g × {})) ↑c 0c) ≠ a1a ( Nc 1(b × {V}) +c Nc 1(g × {})))
121114, 120mpbir 200 . . . . . . 7 (( Nc 1(b × {V}) +c Nc 1(g × {})) ↑c 0c) ≠
12296, 121eqnetrri 2535 . . . . . 6 (( Nc 1b +c Nc 1g) ↑c 0c) ≠
123 addceq12 4385 . . . . . . . 8 ((M = Nc 1b N = Nc 1g) → (M +c N) = ( Nc 1b +c Nc 1g))
124123oveq1d 5537 . . . . . . 7 ((M = Nc 1b N = Nc 1g) → ((M +c N) ↑c 0c) = (( Nc 1b +c Nc 1g) ↑c 0c))
125124neeq1d 2529 . . . . . 6 ((M = Nc 1b N = Nc 1g) → (((M +c N) ↑c 0c) ≠ ↔ (( Nc 1b +c Nc 1g) ↑c 0c) ≠ ))
126122, 125mpbiri 224 . . . . 5 ((M = Nc 1b N = Nc 1g) → ((M +c N) ↑c 0c) ≠ )
12776, 126syl6bir 220 . . . 4 ((M NC N NC ) → ((1b M 1g N) → ((M +c N) ↑c 0c) ≠ ))
128127exlimdvv 1637 . . 3 ((M NC N NC ) → (bg(1b M 1g N) → ((M +c N) ↑c 0c) ≠ ))
12973, 128sylbid 206 . 2 ((M NC N NC ) → (((Mc 0c) ≠ (Nc 0c) ≠ ) → ((M +c N) ↑c 0c) ≠ ))
13068, 129impbid 183 1 ((M NC N NC ) → (((M +c N) ↑c 0c) ≠ ↔ ((Mc 0c) ≠ (Nc 0c) ≠ )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  Vcvv 2859   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  {csn 3737  ℘1cpw1 4135  0cc0c 4374   +c cplc 4375   class class class wbr 4639   × cxp 4770  (class class class)co 5525   ≈ cen 6028   NC cncs 6088   Nc cnc 6091   ↑c cce 6096 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-ce 6106 This theorem is referenced by:  ce0nn  6180
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