Theorem rankprb 9811

Description: The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.)

Assertion

Ref	Expression
rankprb	⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankprb

Step	Hyp	Ref	Expression
1		snwf 9769	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
2		snwf 9769	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → {𝐵} ∈ ∪ (𝑅1 “ On))
3		rankunb 9810	. . . 4 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐵})))
4	1, 2, 3	syl2an 596	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐵})))
5		ranksnb 9787	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))
6		ranksnb 9787	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
7		uneq12 4129	. . . 4 ⊢ (((rank‘{𝐴}) = suc (rank‘𝐴) ∧ (rank‘{𝐵}) = suc (rank‘𝐵)) → ((rank‘{𝐴}) ∪ (rank‘{𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
8	5, 6, 7	syl2an 596	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
9	4, 8	eqtrd 2765	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
10		df-pr 4595	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
11	10	fveq2i 6864	. 2 ⊢ (rank‘{𝐴, 𝐵}) = (rank‘({𝐴} ∪ {𝐵}))
12		rankon 9755	. . . 4 ⊢ (rank‘𝐴) ∈ On
13	12	onordi 6448	. . 3 ⊢ Ord (rank‘𝐴)
14		rankon 9755	. . . 4 ⊢ (rank‘𝐵) ∈ On
15	14	onordi 6448	. . 3 ⊢ Ord (rank‘𝐵)
16		ordsucun 7803	. . 3 ⊢ ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘𝐵)) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
17	13, 15, 16	mp2an 692	. 2 ⊢ suc ((rank‘𝐴) ∪ (rank‘𝐵)) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵))
18	9, 11, 17	3eqtr4g 2790	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3915  {csn 4592  {cpr 4594  ∪ cuni 4874   “ cima 5644  Ord word 6334  Oncon0 6335  suc csuc 6337  ‘cfv 6514  𝑅₁cr1 9722  rankcrnk 9723

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-r1 9724  df-rank 9725

This theorem is referenced by:  rankopb  9812  rankpr  9817  r1limwun  10696  rankaltopb  35974