Theorem rankaltopb 34281

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 9567	. . 3 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → {𝐵} ∈ ∪ (𝑅1 “ On))
2		df-altop 34260	. . . . . 6 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
3	2	fveq2i 6777	. . . . 5 ⊢ (rank‘⟪𝐴, 𝐵⟫) = (rank‘{{𝐴}, {𝐴, {𝐵}}})
4		snwf 9567	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
5	4	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → {𝐴} ∈ ∪ (𝑅1 “ On))
6		prwf 9569	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → {𝐴, {𝐵}} ∈ ∪ (𝑅1 “ On))
7		rankprb 9609	. . . . . 6 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ ∪ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
8	5, 6, 7	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
9	3, 8	eqtrid 2790	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
10		snsspr1 4747	. . . . . . . 8 ⊢ {𝐴} ⊆ {𝐴, {𝐵}}
11		ssequn1 4114	. . . . . . . 8 ⊢ ({𝐴} ⊆ {𝐴, {𝐵}} ↔ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}})
12	10, 11	mpbi 229	. . . . . . 7 ⊢ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}}
13	12	fveq2i 6777	. . . . . 6 ⊢ (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = (rank‘{𝐴, {𝐵}})
14		rankunb 9608	. . . . . . 7 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
15	5, 6, 14	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
16		rankprb 9609	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, {𝐵}}) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
17	13, 15, 16	3eqtr3a 2802	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
18		suceq 6331	. . . . 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 2778	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
21	1, 20	sylan2 593	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
22		ranksnb 9585	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
23	22	uneq2d 4097	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)))
24		suceq 6331	. . . 4 ⊢ (((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
25		suceq 6331	. . . 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 482	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
28	21, 27	eqtrd 2778	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  {csn 4561  {cpr 4563  ∪ cuni 4839   “ cima 5592  Oncon0 6266  suc csuc 6268  ‘cfv 6433  𝑅₁cr1 9520  rankcrnk 9521  ⟪caltop 34258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523  df-altop 34260

This theorem is referenced by: (None)