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Theorem ltfinp1 4462
 Description: One plus a finite cardinal is strictly greater. (Contributed by SF, 29-Jan-2015.)
Assertion
Ref Expression
ltfinp1 ((A V A) → ⟪A, (A +c 1c)⟫ <fin )

Proof of Theorem ltfinp1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . 3 ((A V A) → A)
2 peano1 4402 . . . 4 0c Nn
3 addcid1 4405 . . . . . 6 (A +c 0c) = A
43addceq1i 4386 . . . . 5 ((A +c 0c) +c 1c) = (A +c 1c)
54eqcomi 2357 . . . 4 (A +c 1c) = ((A +c 0c) +c 1c)
6 addceq2 4384 . . . . . . 7 (x = 0c → (A +c x) = (A +c 0c))
76addceq1d 4389 . . . . . 6 (x = 0c → ((A +c x) +c 1c) = ((A +c 0c) +c 1c))
87eqeq2d 2364 . . . . 5 (x = 0c → ((A +c 1c) = ((A +c x) +c 1c) ↔ (A +c 1c) = ((A +c 0c) +c 1c)))
98rspcev 2955 . . . 4 ((0c Nn (A +c 1c) = ((A +c 0c) +c 1c)) → x Nn (A +c 1c) = ((A +c x) +c 1c))
102, 5, 9mp2an 653 . . 3 x Nn (A +c 1c) = ((A +c x) +c 1c)
111, 10jctir 524 . 2 ((A V A) → (A x Nn (A +c 1c) = ((A +c x) +c 1c)))
12 1cex 4142 . . . . 5 1c V
13 addcexg 4393 . . . . 5 ((A V 1c V) → (A +c 1c) V)
1412, 13mpan2 652 . . . 4 (A V → (A +c 1c) V)
15 opkltfing 4449 . . . 4 ((A V (A +c 1c) V) → (⟪A, (A +c 1c)⟫ <fin ↔ (A x Nn (A +c 1c) = ((A +c x) +c 1c))))
1614, 15mpdan 649 . . 3 (A V → (⟪A, (A +c 1c)⟫ <fin ↔ (A x Nn (A +c 1c) = ((A +c x) +c 1c))))
1716adantr 451 . 2 ((A V A) → (⟪A, (A +c 1c)⟫ <fin ↔ (A x Nn (A +c 1c) = ((A +c x) +c 1c))))
1811, 17mpbird 223 1 ((A V A) → ⟪A, (A +c 1c)⟫ <fin )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  Vcvv 2859  ∅c0 3550  ⟪copk 4057  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375
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