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Theorem fin23lem25 9184
Description: Lemma for fin23 9249. In a chain of finite sets, equinumerosity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Assertion
Ref Expression
fin23lem25 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵𝐵𝐴)) → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem fin23lem25
StepHypRef Expression
1 dfpss2 3725 . . . . . . . 8 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 php3 8187 . . . . . . . . . 10 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴𝐵)
3 sdomnen 8026 . . . . . . . . . 10 (𝐴𝐵 → ¬ 𝐴𝐵)
42, 3syl 17 . . . . . . . . 9 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → ¬ 𝐴𝐵)
54ex 449 . . . . . . . 8 (𝐵 ∈ Fin → (𝐴𝐵 → ¬ 𝐴𝐵))
61, 5syl5bir 233 . . . . . . 7 (𝐵 ∈ Fin → ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴𝐵))
76adantl 481 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐴𝐵 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴𝐵))
87expd 451 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵 → (¬ 𝐴 = 𝐵 → ¬ 𝐴𝐵)))
9 dfpss2 3725 . . . . . . . . 9 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐵 = 𝐴))
10 eqcom 2658 . . . . . . . . . . 11 (𝐵 = 𝐴𝐴 = 𝐵)
1110notbii 309 . . . . . . . . . 10 𝐵 = 𝐴 ↔ ¬ 𝐴 = 𝐵)
1211anbi2i 730 . . . . . . . . 9 ((𝐵𝐴 ∧ ¬ 𝐵 = 𝐴) ↔ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵))
139, 12bitri 264 . . . . . . . 8 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐴 = 𝐵))
14 php3 8187 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵𝐴)
15 sdomnen 8026 . . . . . . . . . . 11 (𝐵𝐴 → ¬ 𝐵𝐴)
16 ensym 8046 . . . . . . . . . . 11 (𝐴𝐵𝐵𝐴)
1715, 16nsyl 135 . . . . . . . . . 10 (𝐵𝐴 → ¬ 𝐴𝐵)
1814, 17syl 17 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → ¬ 𝐴𝐵)
1918ex 449 . . . . . . . 8 (𝐴 ∈ Fin → (𝐵𝐴 → ¬ 𝐴𝐵))
2013, 19syl5bir 233 . . . . . . 7 (𝐴 ∈ Fin → ((𝐵𝐴 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴𝐵))
2120adantr 480 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐵𝐴 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴𝐵))
2221expd 451 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐵𝐴 → (¬ 𝐴 = 𝐵 → ¬ 𝐴𝐵)))
238, 22jaod 394 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐴𝐵𝐵𝐴) → (¬ 𝐴 = 𝐵 → ¬ 𝐴𝐵)))
24233impia 1280 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵𝐵𝐴)) → (¬ 𝐴 = 𝐵 → ¬ 𝐴𝐵))
2524con4d 114 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵𝐵𝐴)) → (𝐴𝐵𝐴 = 𝐵))
26 eqeng 8031 . . 3 (𝐴 ∈ Fin → (𝐴 = 𝐵𝐴𝐵))
27263ad2ant1 1102 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵𝐵𝐴)) → (𝐴 = 𝐵𝐴𝐵))
2825, 27impbid 202 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵𝐵𝐴)) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  wpss 3608   class class class wbr 4685  cen 7994  csdm 7996  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001
This theorem is referenced by:  fin23lem23  9186  fin1a2lem9  9268
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