Theorem dif1en 9104

Description: If a set 𝐴 is equinumerous to the successor of an ordinal 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5320. (Revised by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.)

Assertion

Ref	Expression
dif1en	⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1en

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → 𝐴 ≈ suc 𝑀)
2		encv 8891	. . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V))
3	2	simpld 495	. . . 4 ⊢ (𝐴 ≈ suc 𝑀 → 𝐴 ∈ V)
4	3	3anim1i 1152	. . 3 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On))
5		bren 8893	. . . 4 ⊢ (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)
6		sucidg 6398	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
7		f1ocnvdm 7231	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
8	7	3adant2 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
9		f1ofvswap 7252	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ (◡𝑓‘𝑀) ∈ 𝐴) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
10	8, 9	syld3an3 1409	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
11		f1ocnvfv2 7223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝑓‘(◡𝑓‘𝑀)) = 𝑀)
12	11	opeq2d 4837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩ = ⟨𝑋, 𝑀⟩)
13	12	preq1d 4700	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} = {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})
14	13	uneq2d 4123	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) = ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}))
15	14	f1oeq1d 6779	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀))
16	15	3adant2 1131	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀))
17	10, 16	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
18	6, 17	syl3an3 1165	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
19	18	3adant3r1 1182	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀)
20		f1ofun 6786	. . . . . . . . . . 11 ⊢ (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀 → Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}))
21		opex 5421	. . . . . . . . . . . . . 14 ⊢ ⟨𝑋, 𝑀⟩ ∈ V
22	21	prid1 4723	. . . . . . . . . . . . 13 ⊢ ⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}
23		elun2 4137	. . . . . . . . . . . . 13 ⊢ (⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} → ⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}))
24	22, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ ⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})
25		funopfv 6894	. . . . . . . . . . . 12 ⊢ (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) → (⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀))
26	24, 25	mpi 20	. . . . . . . . . . 11 ⊢ (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀)
27	19, 20, 26	3syl 18	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀)
28		simpr2 1195	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → 𝑋 ∈ 𝐴)
29		f1ocnvfv 7224	. . . . . . . . . . 11 ⊢ ((((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀 → (◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀) = 𝑋))
30	19, 28, 29	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑋) = 𝑀 → (◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀) = 𝑋))
31	27, 30	mpd 15	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀) = 𝑋)
32	31	sneqd 4598	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)} = {𝑋})
33	32	difeq2d 4082	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝐴 ∖ {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)}) = (𝐴 ∖ {𝑋}))
34		simpr1 1194	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → 𝐴 ∈ V)
35		3simpc 1150	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On))
36	35	anim2i 617	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)))
37		3anass 1095	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) ↔ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)))
38	36, 37	sylibr 233	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On))
39	34, 38	jca 512	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)))
40		simpl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → 𝐴 ∈ V)
41		simpr3 1196	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → 𝑀 ∈ On)
42	40, 41	jca 512	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝐴 ∈ V ∧ 𝑀 ∈ On))
43		simpr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On))
44	42, 43	jca 512	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → ((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)))
45		vex 3449	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
46	45	resex 5985	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∈ V
47		prex 5389	. . . . . . . . . . 11 ⊢ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} ∈ V
48	46, 47	unex 7680	. . . . . . . . . 10 ⊢ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) ∈ V
49		dif1enlem 9100	. . . . . . . . . 10 ⊢ (((((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)}) ≈ 𝑀)
50	48, 49	mp3anl1 1455	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}):𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)}) ≈ 𝑀)
51	18, 50	sylan2 593	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝐴 ∖ {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)}) ≈ 𝑀)
52	39, 44, 51	3syl 18	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝐴 ∖ {(◡((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩})‘𝑀)}) ≈ 𝑀)
53	33, 52	eqbrtrrd 5129	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
54	53	ex 413	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
55	54	exlimiv 1933	. . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
56	5, 55	sylbi 216	. . 3 ⊢ (𝐴 ≈ suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
57	1, 4, 56	sylc 65	. 2 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
58	57	3comr 1125	1 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908  {csn 4586  {cpr 4588  ⟨cop 4592   class class class wbr 5105  ^◡ccnv 5632   ↾ cres 5635  Oncon0 6317  suc csuc 6319  Fun wfun 6490  –1-1-onto→wf1o 6495  ‘cfv 6496   ≈ cen 8880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-en 8884

This theorem is referenced by:  dif1ennn  9105