Theorem rexdif1en 9169

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 7749. (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 8974	. . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V))
2	1	simpld 500	. . . 4 ⊢ (𝐴 ≈ suc 𝑀 → 𝐴 ∈ V)
3		breng 8975	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ suc 𝑀 ∈ V) → (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀))
4	1, 3	syl 18	. . . . . 6 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀))
5	4	ibi 270	. . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)
6		sucidg 6445	. . . . . . . . . 10 ⊢ (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
7		f1ocnvdm 7291	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
8	7	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑀 ∈ suc 𝑀 ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
9	6, 8	sylan 592	. . . . . . . . 9 ⊢ ((𝑀 ∈ On ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
10	9	adantll 727	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
11		vex 3455	. . . . . . . . 9 ⊢ 𝑓 ∈ V
12		dif1enlem 9168	. . . . . . . . 9 ⊢ (((𝑓 ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀)
13	11, 12	mp3anl1 1484	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀)
14		sneq 4594	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝑓‘𝑀) → {𝑥} = {(◡𝑓‘𝑀)})
15	14	difeq2d 4074	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝑓‘𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(◡𝑓‘𝑀)}))
16	15	breq1d 5113	. . . . . . . . 9 ⊢ (𝑥 = (◡𝑓‘𝑀) → ((𝐴 ∖ {𝑥}) ≈ 𝑀 ↔ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀))
17	16	rspcev 3577	. . . . . . . 8 ⊢ (((◡𝑓‘𝑀) ∈ 𝐴 ∧ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
18	10, 13, 17	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
19	18	ex 418	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (𝑓:𝐴–1-1-onto→suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
20	19	exlimdv 1966	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
21	5, 20	syl5 35	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
22	2, 21	sylan 592	. . 3 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑀 ∈ On) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
23	22	ancoms 464	. 2 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀))
24	23	syldbl2 855	1 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896  {csn 4584   class class class wbr 5103  ^◡ccnv 5650  Oncon0 6361  suc csuc 6363  –1-1-onto→wf1o 6536  ‘cfv 6537   ≈ cen 8963

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-en 8967

This theorem is used by:  findcard2  9173  enp1i  9263