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Theorem onomeneq 9255
Description: An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.) Avoid ax-pow 5361. (Revised by BTernaryTau, 2-Dec-2024.)
Assertion
Ref Expression
onomeneq ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem onomeneq
StepHypRef Expression
1 endom 9002 . . . . . 6 (𝐴𝐵𝐴𝐵)
2 nnfi 9197 . . . . . . . . 9 (𝐵 ∈ ω → 𝐵 ∈ Fin)
3 domfi 9219 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
4 simpr 483 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴𝐵)
53, 4jca 510 . . . . . . . . . 10 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → (𝐴 ∈ Fin ∧ 𝐴𝐵))
6 domnsymfi 9230 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ 𝐴𝐵) → ¬ 𝐵𝐴)
76ex 411 . . . . . . . . . . . . 13 (𝐴 ∈ Fin → (𝐴𝐵 → ¬ 𝐵𝐴))
8 php3 9239 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵𝐴)
98ex 411 . . . . . . . . . . . . 13 (𝐴 ∈ Fin → (𝐵𝐴𝐵𝐴))
107, 9nsyld 156 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (𝐴𝐵 → ¬ 𝐵𝐴))
1110adantl 480 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴 ∈ Fin) → (𝐴𝐵 → ¬ 𝐵𝐴))
1211expimpd 452 . . . . . . . . . 10 (𝐵 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴𝐵) → ¬ 𝐵𝐴))
135, 12syl5 34 . . . . . . . . 9 (𝐵 ∈ ω → ((𝐵 ∈ Fin ∧ 𝐴𝐵) → ¬ 𝐵𝐴))
142, 13mpand 693 . . . . . . . 8 (𝐵 ∈ ω → (𝐴𝐵 → ¬ 𝐵𝐴))
1514adantl 480 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ¬ 𝐵𝐴))
16 eloni 6378 . . . . . . . 8 (𝐴 ∈ On → Ord 𝐴)
17 nnord 7876 . . . . . . . 8 (𝐵 ∈ ω → Ord 𝐵)
18 ordtri1 6401 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
19 ordelpss 6396 . . . . . . . . . . 11 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐵𝐴))
2019ancoms 457 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴𝐵𝐴))
2120notbid 317 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐴))
2218, 21bitrd 278 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
2316, 17, 22syl2an 594 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
2415, 23sylibrd 258 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
251, 24syl5 34 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
26253impia 1114 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴𝐵)
27 ensymfib 9214 . . . . . . . . 9 (𝐵 ∈ Fin → (𝐵𝐴𝐴𝐵))
282, 27syl 17 . . . . . . . 8 (𝐵 ∈ ω → (𝐵𝐴𝐴𝐵))
29 endom 9002 . . . . . . . 8 (𝐵𝐴𝐵𝐴)
3028, 29biimtrrdi 253 . . . . . . 7 (𝐵 ∈ ω → (𝐴𝐵𝐵𝐴))
3130imp 405 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐵𝐴)
32313adant1 1127 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐵𝐴)
33 nndomog 9243 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝐴𝐵𝐴))
3433ancoms 457 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐵𝐴𝐵𝐴))
3534biimp3a 1466 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐵𝐴) → 𝐵𝐴)
3632, 35syld3an3 1406 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐵𝐴)
3726, 36eqssd 3996 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 = 𝐵)
38373expia 1118 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
39 enrefnn 9077 . . . 4 (𝐵 ∈ ω → 𝐵𝐵)
40 breq1 5148 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵𝐵𝐵))
4139, 40syl5ibrcom 246 . . 3 (𝐵 ∈ ω → (𝐴 = 𝐵𝐴𝐵))
4241adantl 480 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
4338, 42impbid 211 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wss 3946  wpss 3947   class class class wbr 5145  Ord word 6367  Oncon0 6368  ωcom 7868  cen 8963  cdom 8964  csdm 8965  Fincfn 8966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-om 7869  df-1o 8488  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970
This theorem is referenced by:  onfin  9257  ficardom  9997  finnisoeu  10149
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