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Theorem rexdif1en 9160
Description: If a set is equinumerous to a nonzero ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals and avoid ax-un 7722. (Revised by BTernaryTau, 5-Jan-2025.)
Assertion
Ref Expression
rexdif1en ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑀

Proof of Theorem rexdif1en
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 encv 8949 . . . . 5 (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V))
21simpld 494 . . . 4 (𝐴 ≈ suc 𝑀𝐴 ∈ V)
3 breng 8950 . . . . . . 7 ((𝐴 ∈ V ∧ suc 𝑀 ∈ V) → (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀))
41, 3syl 17 . . . . . 6 (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀))
54ibi 267 . . . . 5 (𝐴 ≈ suc 𝑀 → ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀)
6 sucidg 6439 . . . . . . . . . 10 (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
7 f1ocnvdm 7279 . . . . . . . . . . 11 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
87ancoms 458 . . . . . . . . . 10 ((𝑀 ∈ suc 𝑀𝑓:𝐴1-1-onto→suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
96, 8sylan 579 . . . . . . . . 9 ((𝑀 ∈ On ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
109adantll 711 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
11 vex 3472 . . . . . . . . 9 𝑓 ∈ V
12 dif1enlem 9158 . . . . . . . . 9 (((𝑓 ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀)
1311, 12mp3anl1 1451 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀)
14 sneq 4633 . . . . . . . . . . 11 (𝑥 = (𝑓𝑀) → {𝑥} = {(𝑓𝑀)})
1514difeq2d 4117 . . . . . . . . . 10 (𝑥 = (𝑓𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(𝑓𝑀)}))
1615breq1d 5151 . . . . . . . . 9 (𝑥 = (𝑓𝑀) → ((𝐴 ∖ {𝑥}) ≈ 𝑀 ↔ (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀))
1716rspcev 3606 . . . . . . . 8 (((𝑓𝑀) ∈ 𝐴 ∧ (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
1810, 13, 17syl2anc 583 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
1918ex 412 . . . . . 6 ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (𝑓:𝐴1-1-onto→suc 𝑀 → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
2019exlimdv 1928 . . . . 5 ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀 → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
215, 20syl5 34 . . . 4 ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (𝐴 ≈ suc 𝑀 → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
222, 21sylan 579 . . 3 ((𝐴 ≈ suc 𝑀𝑀 ∈ On) → (𝐴 ≈ suc 𝑀 → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
2322ancoms 458 . 2 ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑀 → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
2423syldbl2 838 1 ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wrex 3064  Vcvv 3468  cdif 3940  {csn 4623   class class class wbr 5141  ccnv 5668  Oncon0 6358  suc csuc 6360  1-1-ontowf1o 6536  cfv 6537  cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-en 8942
This theorem is referenced by:  findcard2  9166  enp1i  9281
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