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Theorem nnpw1ex 4484
 Description: For any nonempty finite cardinal, there is a unique natural containing a unit power class of one of its elements. Theorem X.1.27 of [Rosser] p. 528. (Contributed by SF, 22-Jan-2015.)
Assertion
Ref Expression
nnpw1ex ((M Nn M) → ∃!n Nn a M 1a n)
Distinct variable group:   n,a,M

Proof of Theorem nnpw1ex
Dummy variables b p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ncfinraise 4481 . . . . . . . . 9 ((M Nn a M a M) → n Nn (1a n 1a n))
2 anidm 625 . . . . . . . . . 10 ((1a n 1a n) ↔ 1a n)
32rexbii 2639 . . . . . . . . 9 (n Nn (1a n 1a n) ↔ n Nn 1a n)
41, 3sylib 188 . . . . . . . 8 ((M Nn a M a M) → n Nn 1a n)
543anidm23 1241 . . . . . . 7 ((M Nn a M) → n Nn 1a n)
65ex 423 . . . . . 6 (M Nn → (a Mn Nn 1a n))
76ancld 536 . . . . 5 (M Nn → (a M → (a M n Nn 1a n)))
87eximdv 1622 . . . 4 (M Nn → (a a Ma(a M n Nn 1a n)))
98imp 418 . . 3 ((M Nn a a M) → a(a M n Nn 1a n))
10 n0 3559 . . . 4 (Ma a M)
1110anbi2i 675 . . 3 ((M Nn M) ↔ (M Nn a a M))
12 rexcom 2772 . . . 4 (n Nn a M 1a na M n Nn 1a n)
13 df-rex 2620 . . . 4 (a M n Nn 1a na(a M n Nn 1a n))
1412, 13bitri 240 . . 3 (n Nn a M 1a na(a M n Nn 1a n))
159, 11, 143imtr4i 257 . 2 ((M Nn M) → n Nn a M 1a n)
16 pw1eq 4143 . . . . . . . 8 (a = b1a = 1b)
1716eleq1d 2419 . . . . . . 7 (a = b → (1a p1b p))
1817cbvrexv 2836 . . . . . 6 (a M 1a pb M 1b p)
1918anbi2i 675 . . . . 5 ((a M 1a n a M 1a p) ↔ (a M 1a n b M 1b p))
20 reeanv 2778 . . . . 5 (a M b M (1a n 1b p) ↔ (a M 1a n b M 1b p))
2119, 20bitr4i 243 . . . 4 ((a M 1a n a M 1a p) ↔ a M b M (1a n 1b p))
22 simplll 734 . . . . . . . 8 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p))) → M Nn )
23 simprll 738 . . . . . . . 8 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p))) → a M)
24 simprlr 739 . . . . . . . 8 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p))) → b M)
25 ncfinraise 4481 . . . . . . . 8 ((M Nn a M b M) → q Nn (1a q 1b q))
2622, 23, 24, 25syl3anc 1182 . . . . . . 7 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p))) → q Nn (1a q 1b q))
27 simp1rl 1020 . . . . . . . . . . . 12 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → n Nn )
28 simp3l 983 . . . . . . . . . . . 12 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → q Nn )
29 simp2rl 1024 . . . . . . . . . . . 12 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → 1a n)
30 simp3rl 1028 . . . . . . . . . . . 12 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → 1a q)
31 nnceleq 4430 . . . . . . . . . . . 12 (((n Nn q Nn ) (1a n 1a q)) → n = q)
3227, 28, 29, 30, 31syl22anc 1183 . . . . . . . . . . 11 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → n = q)
33 simp1rr 1021 . . . . . . . . . . . 12 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → p Nn )
34 simp2rr 1025 . . . . . . . . . . . 12 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → 1b p)
35 simp3rr 1029 . . . . . . . . . . . 12 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → 1b q)
36 nnceleq 4430 . . . . . . . . . . . 12 (((p Nn q Nn ) (1b p 1b q)) → p = q)
3733, 28, 34, 35, 36syl22anc 1183 . . . . . . . . . . 11 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → p = q)
3832, 37eqtr4d 2388 . . . . . . . . . 10 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p)) (q Nn (1a q 1b q))) → n = p)
39383expa 1151 . . . . . . . . 9 (((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p))) (q Nn (1a q 1b q))) → n = p)
4039exp32 588 . . . . . . . 8 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p))) → (q Nn → ((1a q 1b q) → n = p)))
4140rexlimdv 2737 . . . . . . 7 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p))) → (q Nn (1a q 1b q) → n = p))
4226, 41mpd 14 . . . . . 6 ((((M Nn M) (n Nn p Nn )) ((a M b M) (1a n 1b p))) → n = p)
4342exp32 588 . . . . 5 (((M Nn M) (n Nn p Nn )) → ((a M b M) → ((1a n 1b p) → n = p)))
4443rexlimdvv 2744 . . . 4 (((M Nn M) (n Nn p Nn )) → (a M b M (1a n 1b p) → n = p))
4521, 44syl5bi 208 . . 3 (((M Nn M) (n Nn p Nn )) → ((a M 1a n a M 1a p) → n = p))
4645ralrimivva 2706 . 2 ((M Nn M) → n Nn p Nn ((a M 1a n a M 1a p) → n = p))
47 eleq2 2414 . . . 4 (n = p → (1a n1a p))
4847rexbidv 2635 . . 3 (n = p → (a M 1a na M 1a p))
4948reu4 3030 . 2 (∃!n Nn a M 1a n ↔ (n Nn a M 1a n n Nn p Nn ((a M 1a n a M 1a p) → n = p)))
5015, 46, 49sylanbrc 645 1 ((M Nn M) → ∃!n Nn a M 1a n)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616  ∅c0 3550  ℘1cpw1 4135   Nn cnnc 4373 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379 This theorem is referenced by:  tfinprop  4489
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