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Theorem phplem2 9243
Description: Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) Avoid ax-pow 5371. (Revised by BTernaryTau, 4-Nov-2024.)
Hypothesis
Ref Expression
phplem2.1 𝐴 ∈ V
Assertion
Ref Expression
phplem2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵𝐴𝐵))

Proof of Theorem phplem2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bren 8994 . 2 (suc 𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴1-1-onto→suc 𝐵)
2 f1of1 6848 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓:suc 𝐴1-1→suc 𝐵)
3 nnfi 9206 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ Fin)
4 sssucid 6466 . . . . . . . . . 10 𝐴 ⊆ suc 𝐴
5 f1imaenfi 9233 . . . . . . . . . 10 ((𝑓:suc 𝐴1-1→suc 𝐵𝐴 ⊆ suc 𝐴𝐴 ∈ Fin) → (𝑓𝐴) ≈ 𝐴)
64, 5mp3an2 1448 . . . . . . . . 9 ((𝑓:suc 𝐴1-1→suc 𝐵𝐴 ∈ Fin) → (𝑓𝐴) ≈ 𝐴)
72, 3, 6syl2anr 597 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝑓𝐴) ≈ 𝐴)
8 ensymfib 9222 . . . . . . . . . 10 (𝐴 ∈ Fin → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
93, 8syl 17 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
109adantr 480 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝐴 ≈ (𝑓𝐴) ↔ (𝑓𝐴) ≈ 𝐴))
117, 10mpbird 257 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴 ≈ (𝑓𝐴))
12 nnord 7895 . . . . . . . . . 10 (𝐴 ∈ ω → Ord 𝐴)
13 orddif 6482 . . . . . . . . . 10 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
1412, 13syl 17 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
1514imaeq2d 6080 . . . . . . . 8 (𝐴 ∈ ω → (𝑓𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴})))
16 f1ofn 6850 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓 Fn suc 𝐴)
17 phplem2.1 . . . . . . . . . . . 12 𝐴 ∈ V
1817sucid 6468 . . . . . . . . . . 11 𝐴 ∈ suc 𝐴
19 fnsnfv 6988 . . . . . . . . . . 11 ((𝑓 Fn suc 𝐴𝐴 ∈ suc 𝐴) → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
2016, 18, 19sylancl 586 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → {(𝑓𝐴)} = (𝑓 “ {𝐴}))
2120difeq2d 4136 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
22 imadmrn 6090 . . . . . . . . . . . 12 (𝑓 “ dom 𝑓) = ran 𝑓
2322eqcomi 2744 . . . . . . . . . . 11 ran 𝑓 = (𝑓 “ dom 𝑓)
24 f1ofo 6856 . . . . . . . . . . . 12 (𝑓:suc 𝐴1-1-onto→suc 𝐵𝑓:suc 𝐴onto→suc 𝐵)
25 forn 6824 . . . . . . . . . . . 12 (𝑓:suc 𝐴onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
2624, 25syl 17 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → ran 𝑓 = suc 𝐵)
27 f1odm 6853 . . . . . . . . . . . 12 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → dom 𝑓 = suc 𝐴)
2827imaeq2d 6080 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴))
2923, 26, 283eqtr3a 2799 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴))
3029difeq1d 4135 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (suc 𝐵 ∖ {(𝑓𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓𝐴)}))
31 dff1o3 6855 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 ↔ (𝑓:suc 𝐴onto→suc 𝐵 ∧ Fun 𝑓))
32 imadif 6652 . . . . . . . . . 10 (Fun 𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
3331, 32simplbiim 504 . . . . . . . . 9 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴})))
3421, 30, 333eqtr4rd 2786 . . . . . . . 8 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓𝐴)}))
3515, 34sylan9eq 2795 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝑓𝐴) = (suc 𝐵 ∖ {(𝑓𝐴)}))
3611, 35breqtrd 5174 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
37 fnfvelrn 7100 . . . . . . . . . . . 12 ((𝑓 Fn suc 𝐴𝐴 ∈ suc 𝐴) → (𝑓𝐴) ∈ ran 𝑓)
3816, 18, 37sylancl 586 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓𝐴) ∈ ran 𝑓)
3925eleq2d 2825 . . . . . . . . . . . 12 (𝑓:suc 𝐴onto→suc 𝐵 → ((𝑓𝐴) ∈ ran 𝑓 ↔ (𝑓𝐴) ∈ suc 𝐵))
4024, 39syl 17 . . . . . . . . . . 11 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → ((𝑓𝐴) ∈ ran 𝑓 ↔ (𝑓𝐴) ∈ suc 𝐵))
4138, 40mpbid 232 . . . . . . . . . 10 (𝑓:suc 𝐴1-1-onto→suc 𝐵 → (𝑓𝐴) ∈ suc 𝐵)
42 phplem1 9242 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ (𝑓𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
4341, 42sylan2 593 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}))
44 nnfi 9206 . . . . . . . . . . 11 (𝐵 ∈ ω → 𝐵 ∈ Fin)
45 ensymfib 9222 . . . . . . . . . . 11 (𝐵 ∈ Fin → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵))
4644, 45syl 17 . . . . . . . . . 10 (𝐵 ∈ ω → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵))
4746adantr 480 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (𝐵 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ↔ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵))
4843, 47mpbid 232 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵)
49 entrfil 9223 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵) → 𝐴𝐵)
503, 49syl3an1 1162 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓𝐴)}) ≈ 𝐵) → 𝐴𝐵)
5148, 50syl3an3 1164 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)}) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
52513expa 1117 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐴 ≈ (suc 𝐵 ∖ {(𝑓𝐴)})) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
5336, 52syldanl 602 . . . . 5 (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵)) → 𝐴𝐵)
5453anandirs 679 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴1-1-onto→suc 𝐵) → 𝐴𝐵)
5554ex 412 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴1-1-onto→suc 𝐵𝐴𝐵))
5655exlimdv 1931 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑓 𝑓:suc 𝐴1-1-onto→suc 𝐵𝐴𝐵))
571, 56biimtrid 242 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cdif 3960  wss 3963  {csn 4631   class class class wbr 5148  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Ord word 6385  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  ωcom 7887  cen 8981  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-fin 8988
This theorem is referenced by:  nneneq  9244
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