Theorem rankaltopb 32967

Description: Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
rankaltopb	⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))

Proof of Theorem rankaltopb

Step	Hyp	Ref	Expression
1		snwf 9032	. . 3 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → {𝐵} ∈ ∪ (𝑅1 “ On))
2		df-altop 32946	. . . . . 6 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
3	2	fveq2i 6502	. . . . 5 ⊢ (rank‘⟪𝐴, 𝐵⟫) = (rank‘{{𝐴}, {𝐴, {𝐵}}})
4		snwf 9032	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
5	4	adantr 473	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → {𝐴} ∈ ∪ (𝑅1 “ On))
6		prwf 9034	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → {𝐴, {𝐵}} ∈ ∪ (𝑅1 “ On))
7		rankprb 9074	. . . . . 6 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ ∪ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
8	5, 6, 7	syl2anc 576	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
9	3, 8	syl5eq 2826	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
10		snsspr1 4621	. . . . . . . 8 ⊢ {𝐴} ⊆ {𝐴, {𝐵}}
11		ssequn1 4044	. . . . . . . 8 ⊢ ({𝐴} ⊆ {𝐴, {𝐵}} ↔ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}})
12	10, 11	mpbi 222	. . . . . . 7 ⊢ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}}
13	12	fveq2i 6502	. . . . . 6 ⊢ (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = (rank‘{𝐴, {𝐵}})
14		rankunb 9073	. . . . . . 7 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
15	5, 6, 14	syl2anc 576	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
16		rankprb 9074	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, {𝐵}}) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
17	13, 15, 16	3eqtr3a 2838	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
18		suceq 6094	. . . . 5 ⊢ (((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
19	17, 18	syl 17	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
20	9, 19	eqtrd 2814	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
21	1, 20	sylan2 583	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
22		ranksnb 9050	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
23	22	uneq2d 4028	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)))
24		suceq 6094	. . . 4 ⊢ (((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
25		suceq 6094	. . . 4 ⊢ (suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
26	23, 24, 25	3syl 18	. . 3 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
27	26	adantl 474	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
28	21, 27	eqtrd 2814	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ∪ cun 3827   ⊆ wss 3829  {csn 4441  {cpr 4443  ∪ cuni 4712   “ cima 5410  Oncon0 6029  suc csuc 6031  ‘cfv 6188  𝑅₁cr1 8985  rankcrnk 8986  ⟪caltop 32944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-r1 8987  df-rank 8988  df-altop 32946

This theorem is referenced by: (None)