Theorem dif1en 9090

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 5303. (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 1137	. . 3 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → 𝐴 ≈ suc 𝑀)
2		encv 8895	. . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V))
3	2	simpld 494	. . . 4 ⊢ (𝐴 ≈ suc 𝑀 → 𝐴 ∈ V)
4	3	3anim1i 1153	. . 3 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On))
5		bren 8897	. . . 4 ⊢ (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)
6		sucidg 6401	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
7		f1ocnvdm 7234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
8	7	3adant2 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
9		f1ofvswap 7255	. . . . . . . . . . . . . . 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 1412	. . . . . . . . . . . . . 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 7226	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝑓‘(◡𝑓‘𝑀)) = 𝑀)
12	11	opeq2d 4824	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ⟨𝑋, (𝑓‘(◡𝑓‘𝑀))⟩ = ⟨𝑋, 𝑀⟩)
13	12	preq1d 4684	. . . . . . . . . . . . . . . . 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 4109	. . . . . . . . . . . . . . . 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 6770	. . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 1166	. . . . . . . . . . . 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 1184	. . . . . . . . . . 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 6777	. . . . . . . . . . 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 5412	. . . . . . . . . . . . . 14 ⊢ ⟨𝑋, 𝑀⟩ ∈ V
22	21	prid1 4707	. . . . . . . . . . . . 13 ⊢ ⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}
23		elun2 4124	. . . . . . . . . . . . 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 6884	. . . . . . . . . . . 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 1197	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → 𝑋 ∈ 𝐴)
29		f1ocnvfv 7227	. . . . . . . . . . 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 585	. . . . . . . . . 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 4580	. . . . . . . 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 4067	. . . . . . 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 1196	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → 𝐴 ∈ V)
35		3simpc 1151	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On))
36	35	anim2i 618	. . . . . . . . . 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 234	. . . . . . . . 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 511	. . . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → 𝐴 ∈ V)
41		simpr3 1198	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → 𝑀 ∈ On)
42	40, 41	jca 511	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝐴 ∈ V ∧ 𝑀 ∈ On))
43		simpr 484	. . . . . . . . 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 511	. . . . . . . 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 3434	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
46	45	resex 5989	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∈ V
47		prex 5376	. . . . . . . . . . 11 ⊢ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩} ∈ V
48	46, 47	unex 7692	. . . . . . . . . 10 ⊢ ((𝑓 ↾ (𝐴 ∖ {𝑋, (◡𝑓‘𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(◡𝑓‘𝑀), (𝑓‘𝑋)⟩}) ∈ V
49		dif1enlem 9088	. . . . . . . . . 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 1458	. . . . . . . . 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 594	. . . . . . . 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 5110	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On)) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
54	53	ex 412	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
55	54	exlimiv 1932	. . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋 ∈ 𝐴 ∧ 𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
56	5, 55	sylbi 217	. . 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 1126	1 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888  {csn 4568  {cpr 4570  ⟨cop 4574   class class class wbr 5086  ^◡ccnv 5624   ↾ cres 5627  Oncon0 6318  suc csuc 6320  Fun wfun 6487  –1-1-onto→wf1o 6492  ‘cfv 6493   ≈ cen 8884

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-en 8888

This theorem is referenced by:  dif1ennn  9091