Theorem hashunlem 10582

Description: Lemma for hashun 10583. 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 3940	. . . . 5 ⊢ (𝑤 = ∅ → (𝑤 ≈ 𝑗 ↔ ∅ ≈ 𝑗))
2		uneq2 3229	. . . . . 6 ⊢ (𝑤 = ∅ → (𝐴 ∪ 𝑤) = (𝐴 ∪ ∅))
3	2	breq1d 3947	. . . . 5 ⊢ (𝑤 = ∅ → ((𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗)))
4	1, 3	anbi12d 465	. . . 4 ⊢ (𝑤 = ∅ → ((𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ (∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗))))
5	4	rexbidv 2439	. . 3 ⊢ (𝑤 = ∅ → (∃𝑗 ∈ ω (𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ∃𝑗 ∈ ω (∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗))))
6		breq1 3940	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝑗 ↔ 𝑦 ≈ 𝑗))
7		uneq2 3229	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝑦))
8	7	breq1d 3947	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗)))
9	6, 8	anbi12d 465	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))))
10	9	rexbidv 2439	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ω (𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ∃𝑗 ∈ ω (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))))
11		breq1 3940	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ≈ 𝑗 ↔ (𝑦 ∪ {𝑧}) ≈ 𝑗))
12		uneq2 3229	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐴 ∪ 𝑤) = (𝐴 ∪ (𝑦 ∪ {𝑧})))
13	12	breq1d 3947	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗)))
14	11, 13	anbi12d 465	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ((𝑦 ∪ {𝑧}) ≈ 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗))))
15	14	rexbidv 2439	. . 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 3940	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝑤 ≈ 𝑗 ↔ 𝐵 ≈ 𝑗))
17		uneq2 3229	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝐵))
18	17	breq1d 3947	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))
19	16, 18	anbi12d 465	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗))))
20	19	rexbidv 2439	. . 3 ⊢ (𝑤 = 𝐵 → (∃𝑗 ∈ ω (𝑤 ≈ 𝑗 ∧ (𝐴 ∪ 𝑤) ≈ (𝑁 +o 𝑗)) ↔ ∃𝑗 ∈ ω (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗))))
21		peano1 4516	. . . . 5 ⊢ ∅ ∈ ω
22	21	a1i 9	. . . 4 ⊢ (𝜑 → ∅ ∈ ω)
23		0ex 4063	. . . . . 6 ⊢ ∅ ∈ V
24	23	enref 6667	. . . . 5 ⊢ ∅ ≈ ∅
25	24	a1i 9	. . . 4 ⊢ (𝜑 → ∅ ≈ ∅)
26		hashunlem.an	. . . . 5 ⊢ (𝜑 → 𝐴 ≈ 𝑁)
27		un0 3401	. . . . . 6 ⊢ (𝐴 ∪ ∅) = 𝐴
28	27	a1i 9	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ ∅) = 𝐴)
29		hashunlem.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
30		nna0 6378	. . . . . 6 ⊢ (𝑁 ∈ ω → (𝑁 +o ∅) = 𝑁)
31	29, 30	syl 14	. . . . 5 ⊢ (𝜑 → (𝑁 +o ∅) = 𝑁)
32	26, 28, 31	3brtr4d 3968	. . . 4 ⊢ (𝜑 → (𝐴 ∪ ∅) ≈ (𝑁 +o ∅))
33		breq2 3941	. . . . . 6 ⊢ (𝑗 = ∅ → (∅ ≈ 𝑗 ↔ ∅ ≈ ∅))
34		oveq2 5790	. . . . . . 7 ⊢ (𝑗 = ∅ → (𝑁 +o 𝑗) = (𝑁 +o ∅))
35	34	breq2d 3949	. . . . . 6 ⊢ (𝑗 = ∅ → ((𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ ∅) ≈ (𝑁 +o ∅)))
36	33, 35	anbi12d 465	. . . . 5 ⊢ (𝑗 = ∅ → ((∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗)) ↔ (∅ ≈ ∅ ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o ∅))))
37	36	rspcev 2793	. . . 4 ⊢ ((∅ ∈ ω ∧ (∅ ≈ ∅ ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o ∅))) → ∃𝑗 ∈ ω (∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗)))
38	22, 25, 32, 37	syl12anc 1215	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ω (∅ ≈ 𝑗 ∧ (𝐴 ∪ ∅) ≈ (𝑁 +o 𝑗)))
39		peano2 4517	. . . . . . . 8 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
40	39	ad2antlr 481	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → suc 𝑗 ∈ ω)
41		simp-4r 532	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑦 ∈ Fin)
42		vex 2692	. . . . . . . . . 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 522	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → 𝑧 ∈ (𝐵 ∖ 𝑦))
45	44	ad2antrr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑧 ∈ (𝐵 ∖ 𝑦))
46	45	eldifbd 3088	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ¬ 𝑧 ∈ 𝑦)
47	43, 46	eldifd 3086	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑧 ∈ (V ∖ 𝑦))
48		simplr 520	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑗 ∈ ω)
49		simprl 521	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑦 ≈ 𝑗)
50		fiunsnnn 6783	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ 𝑧 ∈ (V ∖ 𝑦)) ∧ (𝑗 ∈ ω ∧ 𝑦 ≈ 𝑗)) → (𝑦 ∪ {𝑧}) ≈ suc 𝑗)
51	41, 47, 48, 49, 50	syl22anc 1218	. . . . . . 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 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝐴 ∈ Fin)
54		simprl 521	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → 𝑦 ⊆ 𝐵)
55	54	ad2antrr 480	. . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∩ 𝐵) = ∅)
58		incom 3273	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝐴) = (𝐴 ∩ 𝑦)
59		incom 3273	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
60	59	eqeq1i 2148	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
61		ssdisj 3424	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐴) = ∅) → (𝑦 ∩ 𝐴) = ∅)
62	60, 61	sylan2b 285	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∩ 𝐴) = ∅)
63	58, 62	syl5eqr 2187	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝑦) = ∅)
64	55, 57, 63	syl2anc 409	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∩ 𝑦) = ∅)
65		unfidisj 6818	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin ∧ (𝐴 ∩ 𝑦) = ∅) → (𝐴 ∪ 𝑦) ∈ Fin)
66	53, 41, 64, 65	syl3anc 1217	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝐴 ∪ 𝑦) ∈ Fin)
67	45	eldifad 3087	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑧 ∈ 𝐵)
68		minel 3429	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ¬ 𝑧 ∈ 𝐴)
69	67, 57, 68	syl2anc 409	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → ¬ 𝑧 ∈ 𝐴)
70		ioran 742	. . . . . . . . . . . 12 ⊢ (¬ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))
71		elun 3222	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
72	70, 71	xchnxbir 671	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))
73	69, 46, 72	sylanbrc 414	. . . . . . . . . 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 3086	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → 𝑁 ∈ ω)
76		nnacl 6384	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (𝑁 +o 𝑗) ∈ ω)
77	75, 48, 76	syl2anc 409	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ 𝑗 ∈ ω) ∧ (𝑦 ≈ 𝑗 ∧ (𝐴 ∪ 𝑦) ≈ (𝑁 +o 𝑗))) → (𝑁 +o 𝑗) ∈ ω)
78		simprr 522	. . . . . . . . 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 6783	. . . . . . . . 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 1218	. . . . . . . 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 3238	. . . . . . . . . 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 2146	. . . . . . . 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 6380	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (𝑁 +o suc 𝑗) = suc (𝑁 +o 𝑗))
85	75, 48, 84	syl2anc 409	. . . . . . . 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 3968	. . . . . . 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 3941	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → ((𝑦 ∪ {𝑧}) ≈ 𝑘 ↔ (𝑦 ∪ {𝑧}) ≈ suc 𝑗))
88		oveq2 5790	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝑁 +o 𝑘) = (𝑁 +o suc 𝑗))
89	88	breq2d 3949	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → ((𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘) ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o suc 𝑗)))
90	87, 89	anbi12d 465	. . . . . . . 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 2793	. . . . . . 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 1215	. . . . . 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 114	. . . . 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 2552	. . . 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 3941	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝑦 ∪ {𝑧}) ≈ 𝑗 ↔ (𝑦 ∪ {𝑧}) ≈ 𝑘))
96		oveq2 5790	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑁 +o 𝑗) = (𝑁 +o 𝑘))
97	96	breq2d 3949	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗) ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘)))
98	95, 97	anbi12d 465	. . . . 5 ⊢ (𝑗 = 𝑘 → (((𝑦 ∪ {𝑧}) ≈ 𝑗 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑗)) ↔ ((𝑦 ∪ {𝑧}) ≈ 𝑘 ∧ (𝐴 ∪ (𝑦 ∪ {𝑧})) ≈ (𝑁 +o 𝑘))))
99	98	cbvrexv 2658	. . . 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	syl6ibr 161	. . 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 6794	. 2 ⊢ (𝜑 → ∃𝑗 ∈ ω (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))
103		simprrr 530	. . 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 6685	. . . . . 6 ⊢ (𝜑 → 𝑀 ≈ 𝐵)
106		simprrl 529	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → 𝐵 ≈ 𝑗)
107		entr 6686	. . . . . 6 ⊢ ((𝑀 ≈ 𝐵 ∧ 𝐵 ≈ 𝑗) → 𝑀 ≈ 𝑗)
108	105, 106, 107	syl2an2r 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → 𝑀 ≈ 𝑗)
109		hashunlem.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ω)
110		simprl 521	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → 𝑗 ∈ ω)
111		nneneq 6759	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝑗 ∈ ω) → (𝑀 ≈ 𝑗 ↔ 𝑀 = 𝑗))
112	109, 110, 111	syl2an2r 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → (𝑀 ≈ 𝑗 ↔ 𝑀 = 𝑗))
113	108, 112	mpbid 146	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → 𝑀 = 𝑗)
114	113	oveq2d 5798	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → (𝑁 +o 𝑀) = (𝑁 +o 𝑗))
115	103, 114	breqtrrd 3964	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ ω ∧ (𝐵 ≈ 𝑗 ∧ (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑗)))) → (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑀))
116	102, 115	rexlimddv 2557	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1332   ∈ wcel 1481  ∃wrex 2418  Vcvv 2689   ∖ cdif 3073   ∪ cun 3074   ∩ cin 3075   ⊆ wss 3076  ∅c0 3368  {csn 3532   class class class wbr 3937  suc csuc 4295  ωcom 4512  (class class class)co 5782   +_o coa 6318   ≈ cen 6640  Fincfn 6642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-fin 6645

This theorem is referenced by:  hashun  10583