Theorem rankopb 9354

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 4757	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	fveq2d 6678	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = (rank‘{{𝐴}, {𝐴, 𝐵}}))
3		snwf 9311	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On))
4		prwf 9313	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → {𝐴, 𝐵} ∈ ∪ (𝑅1 “ On))
5		rankprb 9353	. . 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 4702	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
8		ssequn1 4070	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
9	7, 8	mpbi 233	. . . . 5 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
10	9	fveq2i 6677	. . . 4 ⊢ (rank‘({𝐴} ∪ {𝐴, 𝐵})) = (rank‘{𝐴, 𝐵})
11		rankunb 9352	. . . . 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 9353	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
14	10, 12, 13	3eqtr3a 2797	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
15		suceq 6237	. . 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 2777	1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114   ∪ cun 3841   ⊆ wss 3843  {csn 4516  {cpr 4518  ⟨cop 4522  ∪ cuni 4796   “ cima 5528  Oncon0 6172  suc csuc 6174  ‘cfv 6339  𝑅₁cr1 9264  rankcrnk 9265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-r1 9266  df-rank 9267

This theorem is referenced by:  rankop  9360