Theorem php5 6681

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.

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ (𝑤 = ∅ → 𝑤 = ∅)
2		suceq 4262	. . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅)
3	1, 2	breq12d 3888	. . 3 ⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑤 ↔ ∅ ≈ suc ∅))
4	3	notbid 633	. 2 ⊢ (𝑤 = ∅ → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ ∅ ≈ suc ∅))
5		id 19	. . . 4 ⊢ (𝑤 = 𝑘 → 𝑤 = 𝑘)
6		suceq 4262	. . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘)
7	5, 6	breq12d 3888	. . 3 ⊢ (𝑤 = 𝑘 → (𝑤 ≈ suc 𝑤 ↔ 𝑘 ≈ suc 𝑘))
8	7	notbid 633	. 2 ⊢ (𝑤 = 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝑘 ≈ suc 𝑘))
9		id 19	. . . 4 ⊢ (𝑤 = suc 𝑘 → 𝑤 = suc 𝑘)
10		suceq 4262	. . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘)
11	9, 10	breq12d 3888	. . 3 ⊢ (𝑤 = suc 𝑘 → (𝑤 ≈ suc 𝑤 ↔ suc 𝑘 ≈ suc suc 𝑘))
12	11	notbid 633	. 2 ⊢ (𝑤 = suc 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ suc 𝑘 ≈ suc suc 𝑘))
13		id 19	. . . 4 ⊢ (𝑤 = 𝐴 → 𝑤 = 𝐴)
14		suceq 4262	. . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴)
15	13, 14	breq12d 3888	. . 3 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ suc 𝑤 ↔ 𝐴 ≈ suc 𝐴))
16	15	notbid 633	. 2 ⊢ (𝑤 = 𝐴 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝐴 ≈ suc 𝐴))
17		peano1 4446	. . . . 5 ⊢ ∅ ∈ ω
18		peano3 4448	. . . . 5 ⊢ (∅ ∈ ω → suc ∅ ≠ ∅)
19	17, 18	ax-mp 7	. . . 4 ⊢ suc ∅ ≠ ∅
20		en0 6619	. . . 4 ⊢ (suc ∅ ≈ ∅ ↔ suc ∅ = ∅)
21	19, 20	nemtbir 2356	. . 3 ⊢ ¬ suc ∅ ≈ ∅
22		ensymb 6604	. . 3 ⊢ (suc ∅ ≈ ∅ ↔ ∅ ≈ suc ∅)
23	21, 22	mtbi 636	. 2 ⊢ ¬ ∅ ≈ suc ∅
24		peano2 4447	. . . 4 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
25		vex 2644	. . . . 5 ⊢ 𝑘 ∈ V
26	25	sucex 4353	. . . . 5 ⊢ suc 𝑘 ∈ V
27	25, 26	phplem4 6678	. . . 4 ⊢ ((𝑘 ∈ ω ∧ suc 𝑘 ∈ ω) → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘))
28	24, 27	mpdan 415	. . 3 ⊢ (𝑘 ∈ ω → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘))
29	28	con3d 601	. 2 ⊢ (𝑘 ∈ ω → (¬ 𝑘 ≈ suc 𝑘 → ¬ suc 𝑘 ≈ suc suc 𝑘))
30	4, 8, 12, 16, 23, 29	finds 4452	1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1299   ∈ wcel 1448   ≠ wne 2267  ∅c0 3310   class class class wbr 3875  suc csuc 4225  ωcom 4442   ≈ cen 6562

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-er 6359  df-en 6565

This theorem is referenced by:  snnen2og  6682  1nen2  6684  php5dom  6686  php5fin  6705