Theorem rankopb 9541

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 4799	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	fveq2d 6760	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = (rank‘{{𝐴}, {𝐴, 𝐵}}))
3		snwf 9498	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
4		prwf 9500	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → {𝐴, 𝐵} ∈ ∪ (𝑅1 “ On))
5		rankprb 9540	. . 3 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐴, 𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, 𝐵}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
6	3, 4, 5	syl2an2r 681	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, 𝐵}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
7		snsspr1 4744	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
8		ssequn1 4110	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
9	7, 8	mpbi 229	. . . . 5 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
10	9	fveq2i 6759	. . . 4 ⊢ (rank‘({𝐴} ∪ {𝐴, 𝐵})) = (rank‘{𝐴, 𝐵})
11		rankunb 9539	. . . . 5 ⊢ (({𝐴} ∈ ∪ (𝑅1 “ On) ∧ {𝐴, 𝐵} ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, 𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
12	3, 4, 11	syl2an2r 681	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, 𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
13		rankprb 9540	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
14	10, 12, 13	3eqtr3a 2803	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
15		suceq 6316	. . 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 2782	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∪ cun 3881   ⊆ wss 3883  {csn 4558  {cpr 4560  ⟨cop 4564  ∪ cuni 4836   “ cima 5583  Oncon0 6251  suc csuc 6253  ‘cfv 6418  𝑅₁cr1 9451  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453  df-rank 9454

This theorem is referenced by:  rankop  9547