Theorem hashunlem 10875

Description: Lemma for hashun 10876. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)

Hypotheses

Ref	Expression
hashunlem.a	⊢ (𝜑 → 𝐴 ∈ Fin)
hashunlem.b	⊢ (𝜑 → 𝐵 ∈ Fin)
hashunlem.disj	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
hashunlem.n	⊢ (𝜑 → 𝑁 ∈ ω)
hashunlem.m	⊢ (𝜑 → 𝑀 ∈ ω)
hashunlem.an	⊢ (𝜑 → 𝐴 ≈ 𝑁)
hashunlem.bm	⊢ (𝜑 → 𝐵 ≈ 𝑀)

Assertion

Ref	Expression
hashunlem	⊢ (𝜑 → (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑀))

Proof of Theorem hashunlem

Dummy variables 𝑗 𝑤 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4032	. . . . 5 ⊢ (𝑤 = ∅ → (𝑤 ≈ 𝑗 ↔ ∅ ≈ 𝑗))
2		uneq2 3307	. . . . . 6 ⊢ (𝑤 = ∅ → (𝐴 ∪ 𝑤) = (𝐴 ∪ ∅))
3	2	breq1d 4039	. . . . 5 ⊢ (𝑤 = ∅ → ((𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗)))
4	1, 3	anbi12d 473	. . . 4 ⊢ (𝑤 = ∅ → ((𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ (∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗))))
5	4	rexbidv 2495	. . 3 ⊢ (𝑤 = ∅ → (∃𝑗 ∈ ω (𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ∃𝑗 ∈ ω (∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗))))
6		breq1 4032	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝑗 ↔ 𝑦 ≈ 𝑗))
7		uneq2 3307	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝑦))
8	7	breq1d 4039	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗)))
9	6, 8	anbi12d 473	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))))
10	9	rexbidv 2495	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ω (𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ∃𝑗 ∈ ω (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))))
11		breq1 4032	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ≈ 𝑗 ↔ (𝑦 ∪ {𝑧}) ≈ 𝑗))
12		uneq2 3307	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐴 ∪ 𝑤) = (𝐴 ∪ (𝑦 ∪ {𝑧})))
13	12	breq1d 4039	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗)))
14	11, 13	anbi12d 473	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ((𝑦 ∪ {𝑧}) ≈ 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗))))
15	14	rexbidv 2495	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ω (𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ∃𝑗 ∈ ω ((𝑦 ∪ {𝑧}) ≈ 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗))))
16		breq1 4032	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝑤 ≈ 𝑗 ↔ 𝐵 ≈ 𝑗))
17		uneq2 3307	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝐵))
18	17	breq1d 4039	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))
19	16, 18	anbi12d 473	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗))))
20	19	rexbidv 2495	. . 3 ⊢ (𝑤 = 𝐵 → (∃𝑗 ∈ ω (𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ∃𝑗 ∈ ω (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗))))
21		peano1 4626	. . . . 5 ⊢ ∅ ∈ ω
22	21	a1i 9	. . . 4 ⊢ (𝜑 → ∅ ∈ ω)
23		0ex 4156	. . . . . 6 ⊢ ∅ ∈ V
24	23	enref 6819	. . . . 5 ⊢ ∅ ≈ ∅
25	24	a1i 9	. . . 4 ⊢ (𝜑 → ∅ ≈ ∅)
26		hashunlem.an	. . . . 5 ⊢ (𝜑 → 𝐴 ≈ 𝑁)
27		un0 3480	. . . . . 6 ⊢ (𝐴 ∪ ∅) = 𝐴
28	27	a1i 9	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ ∅) = 𝐴)
29		hashunlem.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
30		nna0 6527	. . . . . 6 ⊢ (𝑁 ∈ ω → (𝑁 +o ∅) = 𝑁)
31	29, 30	syl 14	. . . . 5 ⊢ (𝜑 → (𝑁 +o ∅) = 𝑁)
32	26, 28, 31	3brtr4d 4061	. . . 4 ⊢ (𝜑 → (𝐴 ∪ ∅) ≈ (𝑁 +o ∅))
33		breq2 4033	. . . . . 6 ⊢ (𝑗 = ∅ → (∅ ≈ 𝑗 ↔ ∅ ≈ ∅))
34		oveq2 5926	. . . . . . 7 ⊢ (𝑗 = ∅ → (𝑁 +o 𝑗) = (𝑁 +o ∅))
35	34	breq2d 4041	. . . . . 6 ⊢ (𝑗 = ∅ → ((𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ ∅) ≈ (𝑁 +o ∅)))
36	33, 35	anbi12d 473	. . . . 5 ⊢ (𝑗 = ∅ → ((∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗)) ↔ (∅ ≈ ∅ ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o ∅))))
37	36	rspcev 2864	. . . 4 ⊢ ((∅ ∈ ω ∧ (∅ ≈ ∅ ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o ∅))) → ∃𝑗 ∈ ω (∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗)))
38	22, 25, 32, 37	syl12anc 1247	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ω (∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗)))
39		peano2 4627	. . . . . . . 8 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
40	39	ad2antlr 489	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → suc 𝑗 ∈ ω)
41		simp-4r 542	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑦 ∈ Fin)
42		vex 2763	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
43	42	a1i 9	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑧 ∈ V)
44		simprr 531	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → 𝑧 ∈ (𝐵 ∖ 𝑦))
45	44	ad2antrr 488	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑧 ∈ (𝐵 ∖ 𝑦))
46	45	eldifbd 3165	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ¬ 𝑧 ∈ 𝑦)
47	43, 46	eldifd 3163	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑧 ∈ (V ∖ 𝑦))
48		simplr 528	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑗 ∈ ω)
49		simprl 529	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑦 ≈ 𝑗)
50		fiunsnnn 6937	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ 𝑧 ∈ (V ∖ 𝑦)) ∧ (𝑗 ∈ ω ∧ 𝑦 ≈ 𝑗)) → (𝑦 ∪ {𝑧}) ≈ suc 𝑗)
51	41, 47, 48, 49, 50	syl22anc 1250	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝑦 ∪ {𝑧}) ≈ suc 𝑗)
52		hashunlem.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
53	52	ad4antr 494	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝐴 ∈ Fin)
54		simprl 529	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → 𝑦 ⊆ 𝐵)
55	54	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑦 ⊆ 𝐵)
56		hashunlem.disj	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
57	56	ad4antr 494	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∩ 𝐵) = ∅)
58		incom 3351	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝐴) = (𝐴 ∩ 𝑦)
59		incom 3351	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
60	59	eqeq1i 2201	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
61		ssdisj 3503	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐴) = ∅) → (𝑦 ∩ 𝐴) = ∅)
62	60, 61	sylan2b 287	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∩ 𝐴) = ∅)
63	58, 62	eqtr3id 2240	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝑦) = ∅)
64	55, 57, 63	syl2anc 411	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∩ 𝑦) = ∅)
65		unfidisj 6978	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin ∧ (𝐴 ∩ 𝑦) = ∅) → (𝐴 ∪ 𝑦) ∈ Fin)
66	53, 41, 64, 65	syl3anc 1249	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∪ 𝑦) ∈ Fin)
67	45	eldifad 3164	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑧 ∈ 𝐵)
68		minel 3508	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ¬ 𝑧 ∈ 𝐴)
69	67, 57, 68	syl2anc 411	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ¬ 𝑧 ∈ 𝐴)
70		ioran 753	. . . . . . . . . . . 12 ⊢ (¬ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))
71		elun 3300	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
72	70, 71	xchnxbir 682	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))
73	69, 46, 72	sylanbrc 417	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ¬ 𝑧 ∈ (𝐴 ∪ 𝑦))
74	43, 73	eldifd 3163	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑧 ∈ (V ∖ (𝐴 ∪ 𝑦)))
75	29	ad4antr 494	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑁 ∈ ω)
76		nnacl 6533	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (𝑁 +o 𝑗) ∈ ω)
77	75, 48, 76	syl2anc 411	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝑁 +o 𝑗) ∈ ω)
78		simprr 531	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))
79		fiunsnnn 6937	. . . . . . . . 9 ⊢ ((((𝐴 ∪ 𝑦) ∈ Fin ∧ 𝑧 ∈ (V ∖ (𝐴 ∪ 𝑦))) ∧ ((𝑁 +o 𝑗) ∈ ω ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ suc (𝑁 +o 𝑗))
80	66, 74, 77, 78, 79	syl22anc 1250	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ suc (𝑁 +o 𝑗))
81		unass 3316	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝑦) ∪ {𝑧}) = (𝐴 ∪ (𝑦 ∪ {𝑧}))
82	81	a1i 9	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) = (𝐴 ∪ (𝑦 ∪ {𝑧})))
83	82	eqcomd 2199	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∪ (𝑦 ∪ {𝑧})) = ((𝐴 ∪ 𝑦) ∪ {𝑧}))
84		nnasuc 6529	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (𝑁 +o suc 𝑗) = suc (𝑁 +o 𝑗))
85	75, 48, 84	syl2anc 411	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝑁 +o suc 𝑗) = suc (𝑁 +o 𝑗))
86	80, 83, 85	3brtr4d 4061	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o suc 𝑗))
87		breq2 4033	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → ((𝑦 ∪ {𝑧}) ≈ 𝑘 ↔ (𝑦 ∪ {𝑧}) ≈ suc 𝑗))
88		oveq2 5926	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝑁 +o 𝑘) = (𝑁 +o suc 𝑗))
89	88	breq2d 4041	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → ((𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘) ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o suc 𝑗)))
90	87, 89	anbi12d 473	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → (((𝑦 ∪ {𝑧}) ≈ 𝑘 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘)) ↔ ((𝑦 ∪ {𝑧}) ≈ suc 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o suc 𝑗))))
91	90	rspcev 2864	. . . . . . 7 ⊢ ((suc 𝑗 ∈ ω ∧ ((𝑦 ∪ {𝑧}) ≈ suc 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o suc 𝑗))) → ∃𝑘 ∈ ω ((𝑦 ∪ {𝑧}) ≈ 𝑘 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘)))
92	40, 51, 86, 91	syl12anc 1247	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ∃𝑘 ∈ ω ((𝑦 ∪ {𝑧}) ≈ 𝑘 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘)))
93	92	ex 115	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) → ((𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗)) → ∃𝑘 ∈ ω ((𝑦 ∪ {𝑧}) ≈ 𝑘 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘))))
94	93	rexlimdva 2611	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → (∃𝑗 ∈ ω (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗)) → ∃𝑘 ∈ ω ((𝑦 ∪ {𝑧}) ≈ 𝑘 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘))))
95		breq2 4033	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝑦 ∪ {𝑧}) ≈ 𝑗 ↔ (𝑦 ∪ {𝑧}) ≈ 𝑘))
96		oveq2 5926	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑁 +o 𝑗) = (𝑁 +o 𝑘))
97	96	breq2d 4041	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘)))
98	95, 97	anbi12d 473	. . . . 5 ⊢ (𝑗 = 𝑘 → (((𝑦 ∪ {𝑧}) ≈ 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗)) ↔ ((𝑦 ∪ {𝑧}) ≈ 𝑘 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘))))
99	98	cbvrexv 2727	. . . 4 ⊢ (∃𝑗 ∈ ω ((𝑦 ∪ {𝑧}) ≈ 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗)) ↔ ∃𝑘 ∈ ω ((𝑦 ∪ {𝑧}) ≈ 𝑘 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘)))
100	94, 99	imbitrrdi 162	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → (∃𝑗 ∈ ω (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗)) → ∃𝑗 ∈ ω ((𝑦 ∪ {𝑧}) ≈ 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗))))
101		hashunlem.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
102	5, 10, 15, 20, 38, 100, 101	findcard2sd 6948	. 2 ⊢ (𝜑 → ∃𝑗 ∈ ω (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))
103		simprrr 540	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗))
104		hashunlem.bm	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≈ 𝑀)
105	104	ensymd 6837	. . . . . 6 ⊢ (𝜑 → 𝑀 ≈ 𝐵)
106		simprrl 539	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → 𝐵 ≈ 𝑗)
107		entr 6838	. . . . . 6 ⊢ ((𝑀 ≈ 𝐵 ∧ 𝐵 ≈ 𝑗) → 𝑀 ≈ 𝑗)
108	105, 106, 107	syl2an2r 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → 𝑀 ≈ 𝑗)
109		hashunlem.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ω)
110		simprl 529	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → 𝑗 ∈ ω)
111		nneneq 6913	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝑗 ∈ ω) → (𝑀 ≈ 𝑗 ↔ 𝑀 = 𝑗))
112	109, 110, 111	syl2an2r 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → (𝑀 ≈ 𝑗 ↔ 𝑀 = 𝑗))
113	108, 112	mpbid 147	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → 𝑀 = 𝑗)
114	113	oveq2d 5934	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → (𝑁 +o 𝑀) = (𝑁 +o 𝑗))
115	103, 114	breqtrrd 4057	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑀))
116	102, 115	rexlimddv 2616	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  Vcvv 2760   ∖ cdif 3150   ∪ cun 3151   ∩ cin 3152   ⊆ wss 3153  ∅c0 3446  {csn 3618   class class class wbr 4029  suc csuc 4396  ωcom 4622  (class class class)co 5918   +_o coa 6466   ≈ cen 6792  Fincfn 6794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-fin 6797

This theorem is referenced by:  hashun  10876