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Theorem nneneq 9246
Description: Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.) Avoid ax-pow 5365. (Revised by BTernaryTau, 11-Nov-2024.)
Assertion
Ref Expression
nneneq ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem nneneq
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5146 . . . . . 6 (𝑥 = ∅ → (𝑥𝑧 ↔ ∅ ≈ 𝑧))
2 eqeq1 2741 . . . . . 6 (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
31, 2imbi12d 344 . . . . 5 (𝑥 = ∅ → ((𝑥𝑧𝑥 = 𝑧) ↔ (∅ ≈ 𝑧 → ∅ = 𝑧)))
43ralbidv 3178 . . . 4 (𝑥 = ∅ → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)))
5 breq1 5146 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
6 eqeq1 2741 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
75, 6imbi12d 344 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑧𝑥 = 𝑧) ↔ (𝑦𝑧𝑦 = 𝑧)))
87ralbidv 3178 . . . 4 (𝑥 = 𝑦 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)))
9 breq1 5146 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑧 ↔ suc 𝑦𝑧))
10 eqeq1 2741 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥 = 𝑧 ↔ suc 𝑦 = 𝑧))
119, 10imbi12d 344 . . . . 5 (𝑥 = suc 𝑦 → ((𝑥𝑧𝑥 = 𝑧) ↔ (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
1211ralbidv 3178 . . . 4 (𝑥 = suc 𝑦 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
13 breq1 5146 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑧𝐴𝑧))
14 eqeq1 2741 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑧𝐴 = 𝑧))
1513, 14imbi12d 344 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑧𝑥 = 𝑧) ↔ (𝐴𝑧𝐴 = 𝑧)))
1615ralbidv 3178 . . . 4 (𝑥 = 𝐴 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧)))
17 0fi 9082 . . . . . . 7 ∅ ∈ Fin
18 ensymfib 9224 . . . . . . 7 (∅ ∈ Fin → (∅ ≈ 𝑧𝑧 ≈ ∅))
1917, 18ax-mp 5 . . . . . 6 (∅ ≈ 𝑧𝑧 ≈ ∅)
20 en0 9058 . . . . . . 7 (𝑧 ≈ ∅ ↔ 𝑧 = ∅)
21 eqcom 2744 . . . . . . 7 (𝑧 = ∅ ↔ ∅ = 𝑧)
2220, 21bitri 275 . . . . . 6 (𝑧 ≈ ∅ ↔ ∅ = 𝑧)
2319, 22sylbb 219 . . . . 5 (∅ ≈ 𝑧 → ∅ = 𝑧)
2423rgenw 3065 . . . 4 𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)
25 nn0suc 7916 . . . . . . 7 (𝑤 ∈ ω → (𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧))
26 en0 9058 . . . . . . . . . . . 12 (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)
27 breq2 5147 . . . . . . . . . . . . 13 (𝑤 = ∅ → (suc 𝑦𝑤 ↔ suc 𝑦 ≈ ∅))
28 eqeq2 2749 . . . . . . . . . . . . 13 (𝑤 = ∅ → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = ∅))
2927, 28bibi12d 345 . . . . . . . . . . . 12 (𝑤 = ∅ → ((suc 𝑦𝑤 ↔ suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)))
3026, 29mpbiri 258 . . . . . . . . . . 11 (𝑤 = ∅ → (suc 𝑦𝑤 ↔ suc 𝑦 = 𝑤))
3130biimpd 229 . . . . . . . . . 10 (𝑤 = ∅ → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))
3231a1i 11 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑤 = ∅ → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
33 nfv 1914 . . . . . . . . . . 11 𝑧 𝑦 ∈ ω
34 nfra1 3284 . . . . . . . . . . 11 𝑧𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)
3533, 34nfan 1899 . . . . . . . . . 10 𝑧(𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧))
36 nfv 1914 . . . . . . . . . 10 𝑧(suc 𝑦𝑤 → suc 𝑦 = 𝑤)
37 vex 3484 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
3837phplem2 9245 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 ≈ suc 𝑧𝑦𝑧))
3938imim1d 82 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦𝑧𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧)))
4039ex 412 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (𝑧 ∈ ω → ((𝑦𝑧𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧))))
4140a2d 29 . . . . . . . . . . . . 13 (𝑦 ∈ ω → ((𝑧 ∈ ω → (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧))))
42 rsp 3247 . . . . . . . . . . . . 13 (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑧 ∈ ω → (𝑦𝑧𝑦 = 𝑧)))
4341, 42impel 505 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧)))
44 suceq 6450 . . . . . . . . . . . 12 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
4543, 44syl8 76 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
46 breq2 5147 . . . . . . . . . . . . 13 (𝑤 = suc 𝑧 → (suc 𝑦𝑤 ↔ suc 𝑦 ≈ suc 𝑧))
47 eqeq2 2749 . . . . . . . . . . . . 13 (𝑤 = suc 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = suc 𝑧))
4846, 47imbi12d 344 . . . . . . . . . . . 12 (𝑤 = suc 𝑧 → ((suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
4948biimprcd 250 . . . . . . . . . . 11 ((suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧) → (𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5045, 49syl6 35 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5135, 36, 50rexlimd 3266 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (∃𝑧 ∈ ω 𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5232, 51jaod 860 . . . . . . . 8 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5352ex 412 . . . . . . 7 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5425, 53syl7 74 . . . . . 6 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑤 ∈ ω → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5554ralrimdv 3152 . . . . 5 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ∀𝑤 ∈ ω (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
56 breq2 5147 . . . . . . 7 (𝑤 = 𝑧 → (suc 𝑦𝑤 ↔ suc 𝑦𝑧))
57 eqeq2 2749 . . . . . . 7 (𝑤 = 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = 𝑧))
5856, 57imbi12d 344 . . . . . 6 (𝑤 = 𝑧 → ((suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
5958cbvralvw 3237 . . . . 5 (∀𝑤 ∈ ω (suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧))
6055, 59imbitrdi 251 . . . 4 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
614, 8, 12, 16, 24, 60finds 7918 . . 3 (𝐴 ∈ ω → ∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧))
62 breq2 5147 . . . . 5 (𝑧 = 𝐵 → (𝐴𝑧𝐴𝐵))
63 eqeq2 2749 . . . . 5 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
6462, 63imbi12d 344 . . . 4 (𝑧 = 𝐵 → ((𝐴𝑧𝐴 = 𝑧) ↔ (𝐴𝐵𝐴 = 𝐵)))
6564rspcv 3618 . . 3 (𝐵 ∈ ω → (∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧) → (𝐴𝐵𝐴 = 𝐵)))
6661, 65mpan9 506 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
67 enrefnn 9087 . . . 4 (𝐴 ∈ ω → 𝐴𝐴)
68 breq2 5147 . . . 4 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
6967, 68syl5ibcom 245 . . 3 (𝐴 ∈ ω → (𝐴 = 𝐵𝐴𝐵))
7069adantr 480 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
7166, 70impbid 212 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  wrex 3070  c0 4333   class class class wbr 5143  suc csuc 6386  ωcom 7887  cen 8982  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989
This theorem is referenced by:  php  9247  onomeneqOLD  9266  nnsdomo  9270  fineqvlem  9298  dif1ennnALT  9311  cardnn  10003  satfun  35416
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