Theorem php5 7112

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 4523	. . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅)
3	1, 2	breq12d 4122	. . 3 ⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑤 ↔ ∅ ≈ suc ∅))
4	3	notbid 673	. 2 ⊢ (𝑤 = ∅ → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ ∅ ≈ suc ∅))
5		id 19	. . . 4 ⊢ (𝑤 = 𝑘 → 𝑤 = 𝑘)
6		suceq 4523	. . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘)
7	5, 6	breq12d 4122	. . 3 ⊢ (𝑤 = 𝑘 → (𝑤 ≈ suc 𝑤 ↔ 𝑘 ≈ suc 𝑘))
8	7	notbid 673	. 2 ⊢ (𝑤 = 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝑘 ≈ suc 𝑘))
9		id 19	. . . 4 ⊢ (𝑤 = suc 𝑘 → 𝑤 = suc 𝑘)
10		suceq 4523	. . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘)
11	9, 10	breq12d 4122	. . 3 ⊢ (𝑤 = suc 𝑘 → (𝑤 ≈ suc 𝑤 ↔ suc 𝑘 ≈ suc suc 𝑘))
12	11	notbid 673	. 2 ⊢ (𝑤 = suc 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ suc 𝑘 ≈ suc suc 𝑘))
13		id 19	. . . 4 ⊢ (𝑤 = 𝐴 → 𝑤 = 𝐴)
14		suceq 4523	. . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴)
15	13, 14	breq12d 4122	. . 3 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ suc 𝑤 ↔ 𝐴 ≈ suc 𝐴))
16	15	notbid 673	. 2 ⊢ (𝑤 = 𝐴 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝐴 ≈ suc 𝐴))
17		peano1 4716	. . . . 5 ⊢ ∅ ∈ ω
18		peano3 4718	. . . . 5 ⊢ (∅ ∈ ω → suc ∅ ≠ ∅)
19	17, 18	ax-mp 5	. . . 4 ⊢ suc ∅ ≠ ∅
20		en0 7035	. . . 4 ⊢ (suc ∅ ≈ ∅ ↔ suc ∅ = ∅)
21	19, 20	nemtbir 2501	. . 3 ⊢ ¬ suc ∅ ≈ ∅
22		ensymb 7020	. . 3 ⊢ (suc ∅ ≈ ∅ ↔ ∅ ≈ suc ∅)
23	21, 22	mtbi 677	. 2 ⊢ ¬ ∅ ≈ suc ∅
24		peano2 4717	. . . 4 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
25		vex 2816	. . . . 5 ⊢ 𝑘 ∈ V
26	25	sucex 4621	. . . . 5 ⊢ suc 𝑘 ∈ V
27	25, 26	phplem4 7109	. . . 4 ⊢ ((𝑘 ∈ ω ∧ suc 𝑘 ∈ ω) → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘))
28	24, 27	mpdan 421	. . 3 ⊢ (𝑘 ∈ ω → (suc 𝑘 ≈ suc suc 𝑘 → 𝑘 ≈ suc 𝑘))
29	28	con3d 636	. 2 ⊢ (𝑘 ∈ ω → (¬ 𝑘 ≈ suc 𝑘 → ¬ suc 𝑘 ≈ suc suc 𝑘))
30	4, 8, 12, 16, 23, 29	finds 4722	1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∅c0 3508   class class class wbr 4109  suc csuc 4486  ωcom 4712   ≈ cen 6973

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-er 6767  df-en 6976

This theorem is referenced by:  snnen2og  7113  1nen2  7115  php5dom  7117  php5fin  7139