Theorem rankopb 9737

Description: The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.)

Assertion

Ref	Expression
rankopb	⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankopb

Step	Hyp	Ref	Expression
1		dfopg 4821	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	fveq2d 6821	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = (rank‘{{𝐴}, {𝐴, 𝐵}}))
3		snwf 9694	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
4		prwf 9696	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → {𝐴, 𝐵} ∈ ∪ (𝑅1 “ On))
5		rankprb 9736	. . 3 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐴, 𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, 𝐵}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
6	3, 4, 5	syl2an2r 685	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, 𝐵}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
7		snsspr1 4764	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
8		ssequn1 4134	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
9	7, 8	mpbi 230	. . . . 5 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
10	9	fveq2i 6820	. . . 4 ⊢ (rank‘({𝐴} ∪ {𝐴, 𝐵})) = (rank‘{𝐴, 𝐵})
11		rankunb 9735	. . . . 5 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐴, 𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, 𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
12	3, 4, 11	syl2an2r 685	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, 𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
13		rankprb 9736	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
14	10, 12, 13	3eqtr3a 2789	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
15		suceq 6370	. . 3 ⊢ (((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
16	14, 15	syl 17	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
17	2, 6, 16	3eqtrd 2769	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ∪ cun 3898   ⊆ wss 3900  {csn 4574  {cpr 4576  ⟨cop 4580  ∪ cuni 4857   “ cima 5617  Oncon0 6302  suc csuc 6304  ‘cfv 6477  𝑅₁cr1 9647  rankcrnk 9648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  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 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-r1 9649  df-rank 9650

This theorem is referenced by:  rankop  9743