Theorem php5 7015

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 4492	. . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅)
3	1, 2	breq12d 4095	. . 3 ⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑤 ↔ ∅ ≈ suc ∅))
4	3	notbid 671	. 2 ⊢ (𝑤 = ∅ → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ ∅ ≈ suc ∅))
5		id 19	. . . 4 ⊢ (𝑤 = 𝑘 → 𝑤 = 𝑘)
6		suceq 4492	. . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘)
7	5, 6	breq12d 4095	. . 3 ⊢ (𝑤 = 𝑘 → (𝑤 ≈ suc 𝑤 ↔ 𝑘 ≈ suc 𝑘))
8	7	notbid 671	. 2 ⊢ (𝑤 = 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝑘 ≈ suc 𝑘))
9		id 19	. . . 4 ⊢ (𝑤 = suc 𝑘 → 𝑤 = suc 𝑘)
10		suceq 4492	. . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘)
11	9, 10	breq12d 4095	. . 3 ⊢ (𝑤 = suc 𝑘 → (𝑤 ≈ suc 𝑤 ↔ suc 𝑘 ≈ suc suc 𝑘))
12	11	notbid 671	. 2 ⊢ (𝑤 = suc 𝑘 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ suc 𝑘 ≈ suc suc 𝑘))
13		id 19	. . . 4 ⊢ (𝑤 = 𝐴 → 𝑤 = 𝐴)
14		suceq 4492	. . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴)
15	13, 14	breq12d 4095	. . 3 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ suc 𝑤 ↔ 𝐴 ≈ suc 𝐴))
16	15	notbid 671	. 2 ⊢ (𝑤 = 𝐴 → (¬ 𝑤 ≈ suc 𝑤 ↔ ¬ 𝐴 ≈ suc 𝐴))
17		peano1 4685	. . . . 5 ⊢ ∅ ∈ ω
18		peano3 4687	. . . . 5 ⊢ (∅ ∈ ω → suc ∅ ≠ ∅)
19	17, 18	ax-mp 5	. . . 4 ⊢ suc ∅ ≠ ∅
20		en0 6945	. . . 4 ⊢ (suc ∅ ≈ ∅ ↔ suc ∅ = ∅)
21	19, 20	nemtbir 2489	. . 3 ⊢ ¬ suc ∅ ≈ ∅
22		ensymb 6930	. . 3 ⊢ (suc ∅ ≈ ∅ ↔ ∅ ≈ suc ∅)
23	21, 22	mtbi 674	. 2 ⊢ ¬ ∅ ≈ suc ∅
24		peano2 4686	. . . 4 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
25		vex 2802	. . . . 5 ⊢ 𝑘 ∈ V
26	25	sucex 4590	. . . . 5 ⊢ suc 𝑘 ∈ V
27	25, 26	phplem4 7012	. . . 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 634	. 2 ⊢ (𝑘 ∈ ω → (¬ 𝑘 ≈ suc 𝑘 → ¬ suc 𝑘 ≈ suc suc 𝑘))
30	4, 8, 12, 16, 23, 29	finds 4691	1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∅c0 3491   class class class wbr 4082  suc csuc 4455  ωcom 4681   ≈ cen 6883

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886

This theorem is referenced by:  snnen2og  7016  1nen2  7018  php5dom  7020  php5fin  7040