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Theorem mof0ALT 45744
Description: Alternate proof for mof0 45742 with stronger requirements on distinct variables. Uses mo4 2567. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mof0ALT ∃*𝑓 𝑓:𝐴⟶∅
Distinct variable group:   𝐴,𝑓

Proof of Theorem mof0ALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 f00 6570 . . . . 5 (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
21simplbi 501 . . . 4 (𝑓:𝐴⟶∅ → 𝑓 = ∅)
3 f00 6570 . . . . 5 (𝑔:𝐴⟶∅ ↔ (𝑔 = ∅ ∧ 𝐴 = ∅))
43simplbi 501 . . . 4 (𝑔:𝐴⟶∅ → 𝑔 = ∅)
5 eqtr3 2761 . . . 4 ((𝑓 = ∅ ∧ 𝑔 = ∅) → 𝑓 = 𝑔)
62, 4, 5syl2an 599 . . 3 ((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
76gen2 1803 . 2 𝑓𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔)
8 feq1 6495 . . 3 (𝑓 = 𝑔 → (𝑓:𝐴⟶∅ ↔ 𝑔:𝐴⟶∅))
98mo4 2567 . 2 (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∀𝑓𝑔((𝑓:𝐴⟶∅ ∧ 𝑔:𝐴⟶∅) → 𝑓 = 𝑔))
107, 9mpbir 234 1 ∃*𝑓 𝑓:𝐴⟶∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1540   = wceq 1542  ∃*wmo 2539  c0 4221  wf 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-fun 6351  df-fn 6352  df-f 6353
This theorem is referenced by: (None)
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