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Theorem karden 9312
Description: If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 9966). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9311 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥𝑥𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.)
Hypotheses
Ref Expression
karden.a 𝐴 ∈ V
karden.c 𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
karden.d 𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Assertion
Ref Expression
karden (𝐶 = 𝐷𝐴𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem karden
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 karden.a . . . . . . 7 𝐴 ∈ V
2 breq1 5036 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
31enref 8529 . . . . . . 7 𝐴𝐴
41, 2, 3ceqsexv2d 3493 . . . . . 6 𝑤 𝑤𝐴
5 abn0 4293 . . . . . 6 ({𝑤𝑤𝐴} ≠ ∅ ↔ ∃𝑤 𝑤𝐴)
64, 5mpbir 234 . . . . 5 {𝑤𝑤𝐴} ≠ ∅
7 scott0 9303 . . . . . 6 ({𝑤𝑤𝐴} = ∅ ↔ {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
87necon3bii 3042 . . . . 5 ({𝑤𝑤𝐴} ≠ ∅ ↔ {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅)
96, 8mpbi 233 . . . 4 {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅
10 rabn0 4296 . . . 4 ({𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦))
119, 10mpbi 233 . . 3 𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)
12 vex 3447 . . . . . . . 8 𝑧 ∈ V
13 breq1 5036 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤𝐴𝑧𝐴))
1412, 13elab 3618 . . . . . . 7 (𝑧 ∈ {𝑤𝑤𝐴} ↔ 𝑧𝐴)
15 breq1 5036 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
1615ralab 3635 . . . . . . 7 (∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
1714, 16anbi12i 629 . . . . . 6 ((𝑧 ∈ {𝑤𝑤𝐴} ∧ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
18 simpl 486 . . . . . . . . 9 ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐴)
1918a1i 11 . . . . . . . 8 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐴))
20 karden.c . . . . . . . . . . . 12 𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
21 karden.d . . . . . . . . . . . 12 𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
2220, 21eqeq12i 2816 . . . . . . . . . . 11 (𝐶 = 𝐷 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
23 abbi 2868 . . . . . . . . . . 11 (∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
2422, 23bitr4i 281 . . . . . . . . . 10 (𝐶 = 𝐷 ↔ ∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
25 breq1 5036 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
26 fveq2 6649 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
2726sseq1d 3949 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑦)))
2827imbi2d 344 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
2928albidv 1921 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3025, 29anbi12d 633 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
31 breq1 5036 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
3227imbi2d 344 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3332albidv 1921 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3431, 33anbi12d 633 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
3530, 34bibi12d 349 . . . . . . . . . . 11 (𝑥 = 𝑧 → (((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) ↔ ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))))
3635spvv 2003 . . . . . . . . . 10 (∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
3724, 36sylbi 220 . . . . . . . . 9 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
38 simpl 486 . . . . . . . . 9 ((𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐵)
3937, 38syl6bi 256 . . . . . . . 8 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐵))
4019, 39jcad 516 . . . . . . 7 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧𝐴𝑧𝐵)))
41 ensym 8545 . . . . . . . 8 (𝑧𝐴𝐴𝑧)
42 entr 8548 . . . . . . . 8 ((𝐴𝑧𝑧𝐵) → 𝐴𝐵)
4341, 42sylan 583 . . . . . . 7 ((𝑧𝐴𝑧𝐵) → 𝐴𝐵)
4440, 43syl6 35 . . . . . 6 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝐴𝐵))
4517, 44syl5bi 245 . . . . 5 (𝐶 = 𝐷 → ((𝑧 ∈ {𝑤𝑤𝐴} ∧ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) → 𝐴𝐵))
4645expd 419 . . . 4 (𝐶 = 𝐷 → (𝑧 ∈ {𝑤𝑤𝐴} → (∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴𝐵)))
4746rexlimdv 3245 . . 3 (𝐶 = 𝐷 → (∃𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴𝐵))
4811, 47mpi 20 . 2 (𝐶 = 𝐷𝐴𝐵)
49 enen2 8646 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
50 enen2 8646 . . . . . . 7 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
5150imbi1d 345 . . . . . 6 (𝐴𝐵 → ((𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
5251albidv 1921 . . . . 5 (𝐴𝐵 → (∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
5349, 52anbi12d 633 . . . 4 (𝐴𝐵 → ((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
5453abbidv 2865 . . 3 (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
5554, 20, 213eqtr4g 2861 . 2 (𝐴𝐵𝐶 = 𝐷)
5648, 55impbii 212 1 (𝐶 = 𝐷𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  wss 3884  c0 4246   class class class wbr 5033  cfv 6328  cen 8493  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-r1 9181  df-rank 9182
This theorem is referenced by: (None)
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