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Theorem rankopb 8576
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
StepHypRef Expression
1 dfopg 4332 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21fveq2d 6092 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = (rank‘{{𝐴}, {𝐴, 𝐵}}))
3 snwf 8533 . . . 4 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
43adantr 479 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → {𝐴} ∈ (𝑅1 “ On))
5 prwf 8535 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → {𝐴, 𝐵} ∈ (𝑅1 “ On))
6 rankprb 8575 . . 3 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, 𝐵} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, 𝐵}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
74, 5, 6syl2anc 690 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, 𝐵}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
8 snsspr1 4284 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
9 ssequn1 3744 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
108, 9mpbi 218 . . . . 5 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
1110fveq2i 6091 . . . 4 (rank‘({𝐴} ∪ {𝐴, 𝐵})) = (rank‘{𝐴, 𝐵})
12 rankunb 8574 . . . . 5 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, 𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, 𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
134, 5, 12syl2anc 690 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, 𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
14 rankprb 8575 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
1511, 13, 143eqtr3a 2667 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
16 suceq 5693 . . 3 (((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
1715, 16syl 17 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
182, 7, 173eqtrd 2647 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cun 3537  wss 3539  {csn 4124  {cpr 4126  cop 4130   cuni 4366  cima 5031  Oncon0 5626  suc csuc 5628  cfv 5790  𝑅1cr1 8486  rankcrnk 8487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6936  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-r1 8488  df-rank 8489
This theorem is referenced by:  rankop  8582
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