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Theorem rexdif1en 8763
Description: If a set is equinumerous to a nonzero finite 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.)
Assertion
Ref Expression
rexdif1en ((𝑀 ∈ ω ∧ 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 bren 8567 . 2 (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀)
2 19.42v 1961 . . 3 (∃𝑓(𝑀 ∈ ω ∧ 𝑓:𝐴1-1-onto→suc 𝑀) ↔ (𝑀 ∈ ω ∧ ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀))
3 sucidg 6251 . . . . . 6 (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
4 f1ocnvdm 7055 . . . . . . 7 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
54ancoms 462 . . . . . 6 ((𝑀 ∈ suc 𝑀𝑓:𝐴1-1-onto→suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
63, 5sylan 583 . . . . 5 ((𝑀 ∈ ω ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
7 vex 3403 . . . . . 6 𝑓 ∈ V
8 dif1enlem 8762 . . . . . 6 ((𝑓 ∈ V ∧ 𝑀 ∈ ω ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀)
97, 8mp3an1 1449 . . . . 5 ((𝑀 ∈ ω ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀)
10 sneq 4527 . . . . . . . 8 (𝑥 = (𝑓𝑀) → {𝑥} = {(𝑓𝑀)})
1110difeq2d 4014 . . . . . . 7 (𝑥 = (𝑓𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(𝑓𝑀)}))
1211breq1d 5041 . . . . . 6 (𝑥 = (𝑓𝑀) → ((𝐴 ∖ {𝑥}) ≈ 𝑀 ↔ (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀))
1312rspcev 3527 . . . . 5 (((𝑓𝑀) ∈ 𝐴 ∧ (𝐴 ∖ {(𝑓𝑀)}) ≈ 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
146, 9, 13syl2anc 587 . . . 4 ((𝑀 ∈ ω ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
1514exlimiv 1937 . . 3 (∃𝑓(𝑀 ∈ ω ∧ 𝑓:𝐴1-1-onto→suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
162, 15sylbir 238 . 2 ((𝑀 ∈ ω ∧ ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
171, 16sylan2b 597 1 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wex 1786  wcel 2114  wrex 3055  Vcvv 3399  cdif 3841  {csn 4517   class class class wbr 5031  ccnv 5525  suc csuc 6175  1-1-ontowf1o 6339  cfv 6340  ωcom 7602  cen 8555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-om 7603  df-en 8559
This theorem is referenced by:  findcard2  8766
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