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Theorem cardne 9959
Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
Assertion
Ref Expression
cardne (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
Proof of Theorem cardne
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6921 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐡 ∈ dom card)
2 cardon 9938 . . . . . . . . . 10 (cardβ€˜π΅) ∈ On
32oneli 6471 . . . . . . . . 9 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 ∈ On)
4 breq1 5144 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ (π‘₯ β‰ˆ 𝐡 ↔ 𝐴 β‰ˆ 𝐡))
54onintss 6408 . . . . . . . . 9 (𝐴 ∈ On β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
63, 5syl 17 . . . . . . . 8 (𝐴 ∈ (cardβ€˜π΅) β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
76adantl 481 . . . . . . 7 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
8 cardval3 9946 . . . . . . . . 9 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡})
98sseq1d 4008 . . . . . . . 8 (𝐡 ∈ dom card β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
109adantr 480 . . . . . . 7 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐡} βŠ† 𝐴))
117, 10sylibrd 259 . . . . . 6 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΅) βŠ† 𝐴))
12 ontri1 6391 . . . . . . . 8 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
132, 3, 12sylancr 586 . . . . . . 7 (𝐴 ∈ (cardβ€˜π΅) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1413adantl 481 . . . . . 6 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1511, 14sylibd 238 . . . . 5 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 β‰ˆ 𝐡 β†’ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1615con2d 134 . . . 4 ((𝐡 ∈ dom card ∧ 𝐴 ∈ (cardβ€˜π΅)) β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡))
1716ex 412 . . 3 (𝐡 ∈ dom card β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)))
1817pm2.43d 53 . 2 (𝐡 ∈ dom card β†’ (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡))
191, 18mpcom 38 1 (𝐴 ∈ (cardβ€˜π΅) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098  {crab 3426   βŠ† wss 3943  βˆ© cint 4943   class class class wbr 5141  dom cdm 5669  Oncon0 6357  β€˜cfv 6536   β‰ˆ cen 8935  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-sep 5292  ax-nul 5299  ax-pr 5420
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-rab 3427  df-v 3470  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-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-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-ord 6360  df-on 6361  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-en 8939  df-card 9933
This theorem is referenced by:  carden2b  9961  cardlim  9966  cardsdomelir  9967
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