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Theorem php5 6914
Description: A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
Assertion
Ref Expression
php5 (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴)

Proof of Theorem php5
Dummy variables 𝑤 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4 (𝑤 = ∅ → 𝑤 = ∅)
2 suceq 4433 . . . 4 (𝑤 = ∅ → suc 𝑤 = suc ∅)
31, 2breq12d 4042 . . 3 (𝑤 = ∅ → (𝑤 ≈ suc 𝑤 ↔ ∅ ≈ suc ∅))
43notbid 668 . 2 (𝑤 = ∅ → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ ∅ ≈ suc ∅))
5 id 19 . . . 4 (𝑤 = 𝑘𝑤 = 𝑘)
6 suceq 4433 . . . 4 (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘)
75, 6breq12d 4042 . . 3 (𝑤 = 𝑘 → (𝑤 ≈ suc 𝑤𝑘 ≈ suc 𝑘))
87notbid 668 . 2 (𝑤 = 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝑘 ≈ suc 𝑘))
9 id 19 . . . 4 (𝑤 = suc 𝑘𝑤 = suc 𝑘)
10 suceq 4433 . . . 4 (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘)
119, 10breq12d 4042 . . 3 (𝑤 = suc 𝑘 → (𝑤 ≈ suc 𝑤 ↔ suc 𝑘 ≈ suc suc 𝑘))
1211notbid 668 . 2 (𝑤 = suc 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ suc 𝑘 ≈ suc suc 𝑘))
13 id 19 . . . 4 (𝑤 = 𝐴𝑤 = 𝐴)
14 suceq 4433 . . . 4 (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴)
1513, 14breq12d 4042 . . 3 (𝑤 = 𝐴 → (𝑤 ≈ suc 𝑤𝐴 ≈ suc 𝐴))
1615notbid 668 . 2 (𝑤 = 𝐴 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝐴 ≈ suc 𝐴))
17 peano1 4626 . . . . 5 ∅ ∈ ω
18 peano3 4628 . . . . 5 (∅ ∈ ω → suc ∅ ≠ ∅)
1917, 18ax-mp 5 . . . 4 suc ∅ ≠ ∅
20 en0 6849 . . . 4 (suc ∅ ≈ ∅ ↔ suc ∅ = ∅)
2119, 20nemtbir 2453 . . 3 ¬ suc ∅ ≈ ∅
22 ensymb 6834 . . 3 (suc ∅ ≈ ∅ ↔ ∅ ≈ suc ∅)
2321, 22mtbi 671 . 2 ¬ ∅ ≈ suc ∅
24 peano2 4627 . . . 4 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
25 vex 2763 . . . . 5 𝑘 ∈ V
2625sucex 4531 . . . . 5 suc 𝑘 ∈ V
2725, 26phplem4 6911 . . . 4 ((𝑘 ∈ ω ∧ suc 𝑘 ∈ ω) → (suc 𝑘 ≈ suc suc 𝑘𝑘 ≈ suc 𝑘))
2824, 27mpdan 421 . . 3 (𝑘 ∈ ω → (suc 𝑘 ≈ suc suc 𝑘𝑘 ≈ suc 𝑘))
2928con3d 632 . 2 (𝑘 ∈ ω → (¬ 𝑘 ≈ suc 𝑘 → ¬ suc 𝑘 ≈ suc suc 𝑘))
304, 8, 12, 16, 23, 29finds 4632 1 (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1364  wcel 2164  wne 2364  c0 3446   class class class wbr 4029  suc csuc 4396  ωcom 4622  cen 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-er 6587  df-en 6795
This theorem is referenced by:  snnen2og  6915  1nen2  6917  php5dom  6919  php5fin  6938
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