Theorem rankunb 9821

Description: The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
rankunb	⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankunb

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unwf 9781	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
2		rankval3b 9797	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (rank‘(𝐴 ∪ 𝐵)) = ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦})
3	1, 2	sylbi 220	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦})
4	3	eleq2d 2847	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦}))
5		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
6	5	elintrab 4924	. . . . 5 ⊢ (𝑥 ∈ ∩ {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦} ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦))
7	4, 6	bitrdi 290	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦)))
8		elun 4106	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
9		rankelb 9795	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10		elun1 4134	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ (rank‘𝐴) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
11	9, 10	syl6 36	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
12		rankelb 9795	. . . . . . . . 9 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13		elun2 4135	. . . . . . . . 9 ⊢ ((rank‘𝑥) ∈ (rank‘𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
14	12, 13	syl6 36	. . . . . . . 8 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐵 → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
15	11, 14	jaao 969	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
16	8, 15	biimtrid 245	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (𝐴 ∪ 𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
17	16	ralrimiv 3154	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18		rankon 9766	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
19		rankon 9766	. . . . . . 7 ⊢ (rank‘𝐵) ∈ On
20	18, 19	onun2i 6484	. . . . . 6 ⊢ ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21		eleq2 2850	. . . . . . . . 9 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
22	21	ralbidv 3186	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23		eleq2 2850	. . . . . . . 8 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
24	22, 23	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦) ↔ (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))))
25	24	rspcv 3576	. . . . . 6 ⊢ (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On → (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))))
26	20, 25	ax-mp 5	. . . . 5 ⊢ (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦) → (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
27	17, 26	syl5com 32	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴 ∪ 𝐵)(rank‘𝑥) ∈ 𝑦 → 𝑥 ∈ 𝑦) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
28	7, 27	sylbid 243	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴 ∪ 𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
29	28	ssrdv 3942	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30		ssun1 4130	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
31		rankssb 9819	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ (𝐴 ∪ 𝐵) → (rank‘𝐴) ⊆ (rank‘(𝐴 ∪ 𝐵))))
32	30, 31	mpi 21	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank‘(𝐴 ∪ 𝐵)))
33		ssun2 4131	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
34		rankssb 9819	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (𝐵 ⊆ (𝐴 ∪ 𝐵) → (rank‘𝐵) ⊆ (rank‘(𝐴 ∪ 𝐵))))
35	33, 34	mpi 21	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → (rank‘𝐵) ⊆ (rank‘(𝐴 ∪ 𝐵)))
36	32, 35	unssd 4144	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
37	1, 36	sylbi 220	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
38	29, 37	eqssd 3953	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1568   ∈ wcel 2141  ∀wral 3077  {crab 3414   ∪ cun 3902   ⊆ wss 3904  ∪ cuni 4871  ∩ cint 4911   “ cima 5664  Oncon0 6360  ‘cfv 6536  𝑅₁cr1 9733  rankcrnk 9734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9735  df-rank 9736

This theorem is referenced by:  rankprb  9822  rankopb  9823  rankun  9827  rankaltopb  36425