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Theorem rexdif1en 9199
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 7756. (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 8994 . . . . 5 (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V))
21simpld 494 . . . 4 (𝐴 ≈ suc 𝑀𝐴 ∈ V)
3 breng 8995 . . . . . . 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 6464 . . . . . . . . . 10 (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
7 f1ocnvdm 7306 . . . . . . . . . . 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 580 . . . . . . . . 9 ((𝑀 ∈ On ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
109adantll 714 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
11 vex 3483 . . . . . . . . 9 𝑓 ∈ V
12 dif1enlem 9197 . . . . . . . . 9 (((𝑓 ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀)
1311, 12mp3anl1 1456 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀)
14 sneq 4635 . . . . . . . . . . 11 (𝑥 = (𝑓𝑀) → {𝑥} = {(𝑓𝑀)})
1514difeq2d 4125 . . . . . . . . . 10 (𝑥 = (𝑓𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(𝑓𝑀)}))
1615breq1d 5152 . . . . . . . . 9 (𝑥 = (𝑓𝑀) → ((𝐴 ∖ {𝑥}) ≈ 𝑀 ↔ (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀))
1716rspcev 3621 . . . . . . . 8 (((𝑓𝑀) ∈ 𝐴 ∧ (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
1810, 13, 17syl2anc 584 . . . . . . 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 1932 . . . . 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 580 . . 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 841 1 ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107  wrex 3069  Vcvv 3479  cdif 3947  {csn 4625   class class class wbr 5142  ccnv 5683  Oncon0 6383  suc csuc 6385  1-1-ontowf1o 6559  cfv 6560  cen 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-en 8987
This theorem is referenced by:  findcard2  9205  enp1i  9314
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