Theorem rankprb 9889

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 9847	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
2		snwf 9847	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → {𝐵} ∈ ∪ (𝑅1 “ On))
3		rankunb 9888	. . . 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 9865	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))
6		ranksnb 9865	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
7		uneq12 4173	. . . 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 2775	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
10		df-pr 4634	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
11	10	fveq2i 6910	. 2 ⊢ (rank‘{𝐴, 𝐵}) = (rank‘({𝐴} ∪ {𝐵}))
12		rankon 9833	. . . 4 ⊢ (rank‘𝐴) ∈ On
13	12	onordi 6497	. . 3 ⊢ Ord (rank‘𝐴)
14		rankon 9833	. . . 4 ⊢ (rank‘𝐵) ∈ On
15	14	onordi 6497	. . 3 ⊢ Ord (rank‘𝐵)
16		ordsucun 7845	. . 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 2800	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∪ cun 3961  {csn 4631  {cpr 4633  ∪ cuni 4912   “ cima 5692  Ord word 6385  Oncon0 6386  suc csuc 6388  ‘cfv 6563  𝑅₁cr1 9800  rankcrnk 9801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803

This theorem is referenced by:  rankopb  9890  rankpr  9895  r1limwun  10774  rankaltopb  35961