Theorem rexdif1en 9177

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 7734. (Revised by BTernaryTau, 5-Jan-2025.)

Assertion

Ref	Expression
rexdif1en	⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑀

Proof of Theorem rexdif1en

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		encv 8972	. . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V))
2	1	simpld 494	. . . 4 ⊢ (𝐴 ≈ suc 𝑀 → 𝐴 ∈ V)
3		breng 8973	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ suc 𝑀 ∈ V) → (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀))
4	1, 3	syl 17	. . . . . 6 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀))
5	4	ibi 267	. . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)
6		sucidg 6440	. . . . . . . . . 10 ⊢ (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
7		f1ocnvdm 7283	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
8	7	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑀 ∈ suc 𝑀 ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
9	6, 8	sylan 580	. . . . . . . . 9 ⊢ ((𝑀 ∈ On ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
10	9	adantll 714	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
11		vex 3468	. . . . . . . . 9 ⊢ 𝑓 ∈ V
12		dif1enlem 9175	. . . . . . . . 9 ⊢ (((𝑓 ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀)
13	11, 12	mp3anl1 1457	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀)
14		sneq 4616	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝑓‘𝑀) → {𝑥} = {(◡𝑓‘𝑀)})
15	14	difeq2d 4106	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝑓‘𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(◡𝑓‘𝑀)}))
16	15	breq1d 5134	. . . . . . . . 9 ⊢ (𝑥 = (◡𝑓‘𝑀) → ((𝐴 ∖ {𝑥}) ≈ 𝑀 ↔ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀))
17	16	rspcev 3606	. . . . . . . 8 ⊢ (((◡𝑓‘𝑀) ∈ 𝐴 ∧ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
18	10, 13, 17	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
19	18	ex 412	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (𝑓:𝐴–1-1-onto→suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
20	19	exlimdv 1933	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
21	5, 20	syl5 34	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
22	2, 21	sylan 580	. . 3 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑀 ∈ On) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
23	22	ancoms 458	. 2 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
24	23	syldbl2 841	1 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928  {csn 4606   class class class wbr 5124  ^◡ccnv 5658  Oncon0 6357  suc csuc 6359  –1-1-onto→wf1o 6535  ‘cfv 6536   ≈ cen 8961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-en 8965

This theorem is referenced by:  findcard2  9183  enp1i  9290