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Theorem harval2 9991
Description: An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harval2 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem harval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 harval 9554 . . . . . . 7 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴})
21adantr 480 . . . . . 6 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ (harβ€˜π΄) = {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴})
3 sdomel 9123 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ π‘₯ ∈ On) β†’ (𝑦 β‰Ί π‘₯ β†’ 𝑦 ∈ π‘₯))
4 domsdomtr 9111 . . . . . . . . . . . 12 ((𝑦 β‰Ό 𝐴 ∧ 𝐴 β‰Ί π‘₯) β†’ 𝑦 β‰Ί π‘₯)
53, 4impel 505 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ π‘₯ ∈ On) ∧ (𝑦 β‰Ό 𝐴 ∧ 𝐴 β‰Ί π‘₯)) β†’ 𝑦 ∈ π‘₯)
65an4s 657 . . . . . . . . . 10 (((𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴) ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ 𝑦 ∈ π‘₯)
76ancoms 458 . . . . . . . . 9 (((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) ∧ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴)) β†’ 𝑦 ∈ π‘₯)
873impb 1112 . . . . . . . 8 (((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) ∧ 𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴) β†’ 𝑦 ∈ π‘₯)
98rabssdv 4067 . . . . . . 7 ((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) β†’ {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴} βŠ† π‘₯)
109adantl 481 . . . . . 6 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴} βŠ† π‘₯)
112, 10eqsstrd 4015 . . . . 5 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ (harβ€˜π΄) βŠ† π‘₯)
1211expr 456 . . . 4 ((𝐴 ∈ dom card ∧ π‘₯ ∈ On) β†’ (𝐴 β‰Ί π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯))
1312ralrimiva 3140 . . 3 (𝐴 ∈ dom card β†’ βˆ€π‘₯ ∈ On (𝐴 β‰Ί π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯))
14 ssintrab 4968 . . 3 ((harβ€˜π΄) βŠ† ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} ↔ βˆ€π‘₯ ∈ On (𝐴 β‰Ί π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯))
1513, 14sylibr 233 . 2 (𝐴 ∈ dom card β†’ (harβ€˜π΄) βŠ† ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
16 breq2 5145 . . . 4 (π‘₯ = (harβ€˜π΄) β†’ (𝐴 β‰Ί π‘₯ ↔ 𝐴 β‰Ί (harβ€˜π΄)))
17 harcl 9553 . . . . 5 (harβ€˜π΄) ∈ On
1817a1i 11 . . . 4 (𝐴 ∈ dom card β†’ (harβ€˜π΄) ∈ On)
19 harsdom 9989 . . . 4 (𝐴 ∈ dom card β†’ 𝐴 β‰Ί (harβ€˜π΄))
2016, 18, 19elrabd 3680 . . 3 (𝐴 ∈ dom card β†’ (harβ€˜π΄) ∈ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
21 intss1 4960 . . 3 ((harβ€˜π΄) ∈ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} βŠ† (harβ€˜π΄))
2220, 21syl 17 . 2 (𝐴 ∈ dom card β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯} βŠ† (harβ€˜π΄))
2315, 22eqssd 3994 1 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426   βŠ† wss 3943  βˆ© cint 4943   class class class wbr 5141  dom cdm 5669  Oncon0 6357  β€˜cfv 6536   β‰Ό cdom 8936   β‰Ί csdm 8937  harchar 9550  cardccrd 9929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-oi 9504  df-har 9551  df-card 9933
This theorem is referenced by:  harsucnn  9992  alephnbtwn  10065  harval3  42847  aleph1min  42866
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