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Theorem rankaltopb 34610
Description: Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rankaltopb ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐡 ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜βŸͺ𝐴, 𝐡⟫) = suc suc ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)))

Proof of Theorem rankaltopb
StepHypRef Expression
1 snwf 9750 . . 3 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ {𝐡} ∈ βˆͺ (𝑅1 β€œ On))
2 df-altop 34589 . . . . . 6 βŸͺ𝐴, 𝐡⟫ = {{𝐴}, {𝐴, {𝐡}}}
32fveq2i 6846 . . . . 5 (rankβ€˜βŸͺ𝐴, 𝐡⟫) = (rankβ€˜{{𝐴}, {𝐴, {𝐡}}})
4 snwf 9750 . . . . . . 7 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ {𝐴} ∈ βˆͺ (𝑅1 β€œ On))
54adantr 482 . . . . . 6 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ {𝐴} ∈ βˆͺ (𝑅1 β€œ On))
6 prwf 9752 . . . . . 6 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ {𝐴, {𝐡}} ∈ βˆͺ (𝑅1 β€œ On))
7 rankprb 9792 . . . . . 6 (({𝐴} ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐴, {𝐡}} ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜{{𝐴}, {𝐴, {𝐡}}}) = suc ((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})))
85, 6, 7syl2anc 585 . . . . 5 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜{{𝐴}, {𝐴, {𝐡}}}) = suc ((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})))
93, 8eqtrid 2785 . . . 4 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜βŸͺ𝐴, 𝐡⟫) = suc ((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})))
10 snsspr1 4775 . . . . . . . 8 {𝐴} βŠ† {𝐴, {𝐡}}
11 ssequn1 4141 . . . . . . . 8 ({𝐴} βŠ† {𝐴, {𝐡}} ↔ ({𝐴} βˆͺ {𝐴, {𝐡}}) = {𝐴, {𝐡}})
1210, 11mpbi 229 . . . . . . 7 ({𝐴} βˆͺ {𝐴, {𝐡}}) = {𝐴, {𝐡}}
1312fveq2i 6846 . . . . . 6 (rankβ€˜({𝐴} βˆͺ {𝐴, {𝐡}})) = (rankβ€˜{𝐴, {𝐡}})
14 rankunb 9791 . . . . . . 7 (({𝐴} ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐴, {𝐡}} ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜({𝐴} βˆͺ {𝐴, {𝐡}})) = ((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})))
155, 6, 14syl2anc 585 . . . . . 6 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜({𝐴} βˆͺ {𝐴, {𝐡}})) = ((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})))
16 rankprb 9792 . . . . . 6 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜{𝐴, {𝐡}}) = suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})))
1713, 15, 163eqtr3a 2797 . . . . 5 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ ((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})) = suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})))
18 suceq 6384 . . . . 5 (((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})) = suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) β†’ suc ((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})) = suc suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})))
1917, 18syl 17 . . . 4 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ suc ((rankβ€˜{𝐴}) βˆͺ (rankβ€˜{𝐴, {𝐡}})) = suc suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})))
209, 19eqtrd 2773 . . 3 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜βŸͺ𝐴, 𝐡⟫) = suc suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})))
211, 20sylan2 594 . 2 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐡 ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜βŸͺ𝐴, 𝐡⟫) = suc suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})))
22 ranksnb 9768 . . . . 5 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ (rankβ€˜{𝐡}) = suc (rankβ€˜π΅))
2322uneq2d 4124 . . . 4 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) = ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)))
24 suceq 6384 . . . 4 (((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) = ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)) β†’ suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) = suc ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)))
25 suceq 6384 . . . 4 (suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) = suc ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)) β†’ suc suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) = suc suc ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)))
2623, 24, 253syl 18 . . 3 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ suc suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) = suc suc ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)))
2726adantl 483 . 2 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐡 ∈ βˆͺ (𝑅1 β€œ On)) β†’ suc suc ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) = suc suc ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)))
2821, 27eqtrd 2773 1 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐡 ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜βŸͺ𝐴, 𝐡⟫) = suc suc ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3909   βŠ† wss 3911  {csn 4587  {cpr 4589  βˆͺ cuni 4866   β€œ cima 5637  Oncon0 6318  suc csuc 6320  β€˜cfv 6497  π‘…1cr1 9703  rankcrnk 9704  βŸͺcaltop 34587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-r1 9705  df-rank 9706  df-altop 34589
This theorem is referenced by: (None)
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