Theorem rankaltopb 35967

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 9762	. . 3 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → {𝐵} ∈ ∪ (𝑅1 “ On))
2		df-altop 35946	. . . . . 6 ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
3	2	fveq2i 6861	. . . . 5 ⊢ (rank‘⟪𝐴, 𝐵⟫) = (rank‘{{𝐴}, {𝐴, {𝐵}}})
4		snwf 9762	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
5	4	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → {𝐴} ∈ ∪ (𝑅1 “ On))
6		prwf 9764	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → {𝐴, {𝐵}} ∈ ∪ (𝑅1 “ On))
7		rankprb 9804	. . . . . 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 2776	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
10		snsspr1 4778	. . . . . . . 8 ⊢ {𝐴} ⊆ {𝐴, {𝐵}}
11		ssequn1 4149	. . . . . . . 8 ⊢ ({𝐴} ⊆ {𝐴, {𝐵}} ↔ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}})
12	10, 11	mpbi 230	. . . . . . 7 ⊢ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}}
13	12	fveq2i 6861	. . . . . 6 ⊢ (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = (rank‘{𝐴, {𝐵}})
14		rankunb 9803	. . . . . . 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 9804	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, {𝐵}}) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
17	13, 15, 16	3eqtr3a 2788	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
18		suceq 6400	. . . . 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 2764	. . 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 9780	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
23	22	uneq2d 4131	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)))
24		suceq 6400	. . . 4 ⊢ (((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
25		suceq 6400	. . . 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 481	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
28	21, 27	eqtrd 2764	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3912   ⊆ wss 3914  {csn 4589  {cpr 4591  ∪ cuni 4871   “ cima 5641  Oncon0 6332  suc csuc 6334  ‘cfv 6511  𝑅₁cr1 9715  rankcrnk 9716  ⟪caltop 35944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-r1 9717  df-rank 9718  df-altop 35946

This theorem is referenced by: (None)