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Theorem rankaltopb 35255
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 9806 . . 3 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ {𝐡} ∈ βˆͺ (𝑅1 β€œ On))
2 df-altop 35234 . . . . . 6 βŸͺ𝐴, 𝐡⟫ = {{𝐴}, {𝐴, {𝐡}}}
32fveq2i 6893 . . . . 5 (rankβ€˜βŸͺ𝐴, 𝐡⟫) = (rankβ€˜{{𝐴}, {𝐴, {𝐡}}})
4 snwf 9806 . . . . . . 7 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ {𝐴} ∈ βˆͺ (𝑅1 β€œ On))
54adantr 479 . . . . . 6 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ {𝐴} ∈ βˆͺ (𝑅1 β€œ On))
6 prwf 9808 . . . . . 6 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ {𝐡} ∈ βˆͺ (𝑅1 β€œ On)) β†’ {𝐴, {𝐡}} ∈ βˆͺ (𝑅1 β€œ On))
7 rankprb 9848 . . . . . 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 582 . . . . 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 2782 . . . 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 4816 . . . . . . . 8 {𝐴} βŠ† {𝐴, {𝐡}}
11 ssequn1 4179 . . . . . . . 8 ({𝐴} βŠ† {𝐴, {𝐡}} ↔ ({𝐴} βˆͺ {𝐴, {𝐡}}) = {𝐴, {𝐡}})
1210, 11mpbi 229 . . . . . . 7 ({𝐴} βˆͺ {𝐴, {𝐡}}) = {𝐴, {𝐡}}
1312fveq2i 6893 . . . . . 6 (rankβ€˜({𝐴} βˆͺ {𝐴, {𝐡}})) = (rankβ€˜{𝐴, {𝐡}})
14 rankunb 9847 . . . . . . 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 582 . . . . . 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 9848 . . . . . 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 2794 . . . . 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 6429 . . . . 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 2770 . . 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 591 . 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 9824 . . . . 5 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ (rankβ€˜{𝐡}) = suc (rankβ€˜π΅))
2322uneq2d 4162 . . . 4 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ ((rankβ€˜π΄) βˆͺ (rankβ€˜{𝐡})) = ((rankβ€˜π΄) βˆͺ suc (rankβ€˜π΅)))
24 suceq 6429 . . . 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 6429 . . . 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 480 . 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 2770 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 394   = wceq 1539   ∈ wcel 2104   βˆͺ cun 3945   βŠ† wss 3947  {csn 4627  {cpr 4629  βˆͺ cuni 4907   β€œ cima 5678  Oncon0 6363  suc csuc 6365  β€˜cfv 6542  π‘…1cr1 9759  rankcrnk 9760  βŸͺcaltop 35232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-r1 9761  df-rank 9762  df-altop 35234
This theorem is referenced by: (None)
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