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Theorem harval2 9938
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 9501 . . . . . . 7 (𝐴 ∈ dom card β†’ (harβ€˜π΄) = {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴})
21adantr 482 . . . . . 6 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ (harβ€˜π΄) = {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴})
3 sdomel 9071 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ π‘₯ ∈ On) β†’ (𝑦 β‰Ί π‘₯ β†’ 𝑦 ∈ π‘₯))
4 domsdomtr 9059 . . . . . . . . . . . 12 ((𝑦 β‰Ό 𝐴 ∧ 𝐴 β‰Ί π‘₯) β†’ 𝑦 β‰Ί π‘₯)
53, 4impel 507 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ π‘₯ ∈ On) ∧ (𝑦 β‰Ό 𝐴 ∧ 𝐴 β‰Ί π‘₯)) β†’ 𝑦 ∈ π‘₯)
65an4s 659 . . . . . . . . . 10 (((𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴) ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ 𝑦 ∈ π‘₯)
76ancoms 460 . . . . . . . . 9 (((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) ∧ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴)) β†’ 𝑦 ∈ π‘₯)
873impb 1116 . . . . . . . 8 (((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) ∧ 𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴) β†’ 𝑦 ∈ π‘₯)
98rabssdv 4033 . . . . . . 7 ((π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯) β†’ {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴} βŠ† π‘₯)
109adantl 483 . . . . . 6 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝐴} βŠ† π‘₯)
112, 10eqsstrd 3983 . . . . 5 ((𝐴 ∈ dom card ∧ (π‘₯ ∈ On ∧ 𝐴 β‰Ί π‘₯)) β†’ (harβ€˜π΄) βŠ† π‘₯)
1211expr 458 . . . 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 4933 . . 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 5110 . . . 4 (π‘₯ = (harβ€˜π΄) β†’ (𝐴 β‰Ί π‘₯ ↔ 𝐴 β‰Ί (harβ€˜π΄)))
17 harcl 9500 . . . . 5 (harβ€˜π΄) ∈ On
1817a1i 11 . . . 4 (𝐴 ∈ dom card β†’ (harβ€˜π΄) ∈ On)
19 harsdom 9936 . . . 4 (𝐴 ∈ dom card β†’ 𝐴 β‰Ί (harβ€˜π΄))
2016, 18, 19elrabd 3648 . . 3 (𝐴 ∈ dom card β†’ (harβ€˜π΄) ∈ {π‘₯ ∈ On ∣ 𝐴 β‰Ί π‘₯})
21 intss1 4925 . . 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 3962 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   βŠ† wss 3911  βˆ© cint 4908   class class class wbr 5106  dom cdm 5634  Oncon0 6318  β€˜cfv 6497   β‰Ό cdom 8884   β‰Ί csdm 8885  harchar 9497  cardccrd 9876
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-rep 5243  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-rmo 3352  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-se 5590  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-isom 6506  df-riota 7314  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-oi 9451  df-har 9498  df-card 9880
This theorem is referenced by:  harsucnn  9939  alephnbtwn  10012  harval3  41898  aleph1min  41917
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